TPTP Problem File: NUM923

TPTP Problem File: NUM923_40.p

%------------------------------------------------------------------------------
% File     : NUM923_40 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Number Theory
% Problem  : Sum of two squares line 23, 5000 axioms selected
% Version  : NUM923_4 with the conjecture removed
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [TPTP]
% Names    : 

% Status   : Satisfiable
% Rating   : 1.00 v8.2.0
% Syntax   : Number of formulae    : 7091 (2207 unt;1671 typ;   0 def)
%            Number of atoms       : 11770 (4284 equ)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 7428 (1078   ~; 237   |; 689   &)
%                                         (1004 <=>;4420  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   4 avg)
%            Maximal term depth    :   21 (   2 avg)
%            Number of types       :  545 ( 544 usr)
%            Number of type conns  : 1257 ( 688   >; 569   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    : 1126 (1126 usr; 439 con; 0-8 aty)
%            Number of variables   : 11606 (11327   !; 279   ?;11606   :)
% SPC      : TF0_SAT_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 18:41:08
%------------------------------------------------------------------------------
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__In,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__HOL__Obool,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc_,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc___205,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc___206,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Nume,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Nume_207,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Nume_208,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Rat__Orat_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__HOL__Obool,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int__Oint_,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int__Oint__209,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int__Oint__210,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int__Oint__211,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_Itc__p,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_Itc__p_212,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_Itc__p_213,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_Itc__p_214,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__prod_Itc__,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__prod_Itc___215,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__prod_Itc___216,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__prod_Itc___217,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__prod_Itc___218,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__Int__Oint,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__HOL__Obool_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__prod_Itc___223,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__Nat__Onat,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__prod_Itc_,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickche,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickche_224,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickche_225,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__HOL__Oboo,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__RealDef__,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_Itc___226,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc___227,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc___228,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc_,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__229,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__230,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc_,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__231,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__232,type,
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tff(ty_ty_tc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Numeral__Ocod,type,
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tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
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tff(ty_ty_tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
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tff(ty_ty_tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
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tff(ty_ty_tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickcheck____Nar,type,
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tff(ty_ty_tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
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tff(ty_ty_tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__Int__Oint,type,
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tff(ty_ty_tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__Int__Oin,type,
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tff(ty_ty_tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__Nat__Ona,type,
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%----Explicit typings (1127)
tff(sy_c_Archimedean__Field_Oceiling_000tc__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Int__Oint_000tc__In,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__In,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__Na,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__Re,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Int__Oint_000tc__,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Nat__Onat_000tc__,type,
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tff(sy_c_BijectionRel_ObijR_000tc__Int__Oint_000tc__Int__Oint,type,
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tff(sy_c_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDe,type,
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tff(sy_c_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint,type,
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tff(sy_c_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__prod_Itc__Int__Oint_Mtc__In,type,
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tff(sy_c_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat,type,
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tff(sy_c_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000_258,type,
    cOMBC_1801450329t_bool: fun_fu1175943503t_bool ).

tff(sy_c_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000t,type,
    cOMBC_1329014627t_real: fun_fu2073566759t_real ).

tff(sy_c_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Or,type,
    cOMBC_467724138l_real: fun_fu1523905537l_real ).

tff(sy_c_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__H,type,
    cOMBC_33652895t_bool: fun_fu1940123285t_bool ).

tff(sy_c_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__I,type,
    cOMBC_1417754532nt_int: fun_fu179687707nt_int ).

tff(sy_c_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc_,type,
    cOMBC_2037038898nt_int: fun_fu1223580849nt_int ).

tff(sy_c_COMBI_000tc__Int__Oint,type,
    cOMBI_int: fun_int_int ).

tff(sy_c_COMBI_000tc__Nat__Onat,type,
    cOMBI_nat: fun_nat_nat ).

tff(sy_c_COMBI_000tc__RealDef__Oreal,type,
    cOMBI_real: fun_real_real ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Nat__Onat,type,
    cOMBK_bool_nat: bool > fun_nat_bool ).

tff(sy_c_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    cOMBK_1643584600t_bool: nat > fun_fun_int_bool_nat ).

tff(sy_c_COMBK_000tc__Rat__Orat_000tc__Nat__Onat,type,
    cOMBK_rat_nat: rat > fun_nat_rat ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_int_bool_bool: fun_fu587508068t_bool ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod,type,
    cOMBS_2011279539nt_int: fun_fu1890158468nt_int ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__f,type,
    cOMBS_1720984575t_bool: fun_fu1359497956t_bool ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_M,type,
    cOMBS_1470252563nt_int: fun_fu1004947300nt_int ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__f,type,
    cOMBS_1640193733nt_int: fun_fu1594210934nt_int ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__p,type,
    cOMBS_1891347093nt_int: fun_fu1280007140nt_int ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_nat_bool_bool: fun_fu1901564496t_bool ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat,type,
    cOMBS_nat_nat_nat: fun_fu1382375754at_nat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat,type,
    cOMBS_nat_rat_rat: fun_fu941266250at_rat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocomplex,type,
    cOMBS_696648398omplex: fun_fu1913390252omplex ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    cOMBS_nat_real_real: fun_fu247819600t_real ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal,type,
    cOMBS_337378985l_real: fun_fu385122018l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_real_bool_bool: fun_fu1882383245l_bool ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    cOMBS_real_real_real: fun_fu647765353l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef_,type,
    cOMBS_511527878l_real: fun_fu1698495721l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Or,type,
    cOMBS_193881978l_real: fun_fu1613343649l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__R,type,
    cOMBS_107495110l_real: fun_fu1574854795l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__R_259,type,
    cOMBS_163651580l_real: fun_fu1943305629l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__O,type,
    cOMBS_794971624l_real: fun_fu209750539l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__O_260,type,
    cOMBS_902912708l_real: fun_fu2079587053l_real ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool_000tc__,type,
    cOMBS_400904860l_bool: fun_fu1257888773t_bool ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__H,type,
    cOMBS_480885903t_bool: fun_fu581830891t_bool ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__I,type,
    cOMBS_2019229620nt_int: fun_fu1660428975nt_int ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__R,type,
    cOMBS_549334880nt_rat: fun_fu1887196079nt_rat ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc_,type,
    cOMBS_1610675060t_bool: fun_fu620454517t_bool ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc__261,type,
    cOMBS_1676841879t_bool: fun_fu2062813463t_bool ).

tff(sy_c_Code__Numeral_OSuc__code__numeral,type,
    code_S1047413653umeral: code_code_numeral > code_code_numeral ).

tff(sy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__size,type,
    code_c271388182l_size: code_code_numeral > nat ).

tff(sy_c_Code__Numeral_Odiv__mod__code__numeral,type,
    code_d418564891umeral: ( code_code_numeral * code_code_numeral ) > produc1359518119umeral ).

tff(sy_c_Code__Numeral_Oint__of,type,
    code_int_of: code_code_numeral > int ).

tff(sy_c_Code__Numeral_Onat__of__aux,type,
    code_nat_of_aux: code_code_numeral > fun_nat_nat ).

tff(sy_c_Complete__Lattice_OSup__class_OSup_000tc__RealDef__Oreal,type,
    comple124823625p_real: fun_real_bool > real ).

tff(sy_c_Complete__Lattice_OSup__class_OSup_000tc__fun_Itc__Int__Oint_Mtc__HOL__Oboo,type,
    comple219730294t_bool: fun_fu819804446l_bool > fun_int_bool ).

tff(sy_c_Complex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Oarg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: fun_complex_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex_1: fun_re398676606omplex ).

tff(sy_c_Complex_Ocomplex_Ocomplex__size,type,
    complex_size: complex > nat ).

tff(sy_c_Complex_Oexpi,type,
    expi: complex > complex ).

tff(sy_c_Complex_Oii,type,
    ii: complex ).

tff(sy_c_Complex_Orcis,type,
    rcis: ( real * real ) > complex ).

tff(sy_c_Deriv_OBolzano__bisect,type,
    bolzano_bisect: ( fun_Pr1127657839l_bool * real * real ) > fun_na1745536976l_real ).

tff(sy_c_Deriv_Oderiv_000tc__RealDef__Oreal,type,
    deriv_real: ( fun_real_real * real ) > fun_real_bool ).

tff(sy_c_Divides_Oadjust,type,
    adjust: int > fun_Pr241715507nt_int ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Code____Numeral__Ocode____numeral,type,
    div_di1218280263umeral: fun_Co1323339824umeral ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Int__Oint,type,
    div_div_int: fun_int_fun_int_int ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Nat__Onat,type,
    div_div_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Quickcheck____Narrowing__Ocode____int,type,
    div_di1430059507de_int: fun_Qu64793220de_int ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Code____Numeral__Ocode____numeral,type,
    div_mo1740067990umeral: code_code_numeral > fun_Co536835517umeral ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Int__Oint,type,
    div_mod_int: int > fun_int_int ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Nat__Onat,type,
    div_mod_nat: nat > fun_nat_nat ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Quickcheck____Narrowing__Ocode____int,type,
    div_mo231679042de_int: quickcheck_code_int > fun_Qu812286141de_int ).

tff(sy_c_Divides_Odivmod__int,type,
    divmod_int: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_Odivmod__int__rel,type,
    divmod_int_rel: ( int * int ) > fun_Pr974702441t_bool ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Divides_Odivmod__nat__rel,type,
    divmod_nat_rel: ( nat * nat ) > fun_Pr1927655017t_bool ).

tff(sy_c_Divides_OnegDivAlg,type,
    negDivAlg: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_OnegDivAlg__rel,type,
    negDivAlg_rel: fun_Pr313471731t_bool ).

tff(sy_c_Divides_OnegateSnd,type,
    negateSnd: fun_Pr241715507nt_int ).

tff(sy_c_Divides_Opdivmod,type,
    pdivmod: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_OposDivAlg,type,
    posDivAlg: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_OposDivAlg__rel,type,
    posDivAlg_rel: fun_Pr313471731t_bool ).

tff(sy_c_EulerFermat_OBnorRset,type,
    bnorRset: int > fun_int_fun_int_bool ).

tff(sy_c_EulerFermat_ORRset2norRR,type,
    rRset2norRR: ( fun_int_bool * int ) > fun_int_int ).

tff(sy_c_EulerFermat_ORsetR,type,
    rsetR: int > fun_fu819804446l_bool ).

tff(sy_c_EulerFermat_Ois__RRset,type,
    is_RRset: fun_int_bool > fun_int_bool ).

tff(sy_c_EulerFermat_OnoXRRset,type,
    noXRRset: int > fun_int_fun_int_bool ).

tff(sy_c_EulerFermat_OnorRRset,type,
    norRRset: fun_int_fun_int_bool ).

tff(sy_c_EulerFermat_Ophi,type,
    phi: int > nat ).

tff(sy_c_EulerFermat_Ozcongm,type,
    zcongm: int > fun_int_fun_int_bool ).

tff(sy_c_Euler_OMultInvPair,type,
    multInvPair: ( int * int ) > fun_int_fun_int_bool ).

tff(sy_c_Euler_OSetS,type,
    setS: ( int * int ) > fun_fu819804446l_bool ).

tff(sy_c_EvenOdd_OzEven,type,
    zEven: fun_int_bool ).

tff(sy_c_EvenOdd_OzOdd,type,
    zOdd: fun_int_bool ).

tff(sy_c_Fact_Ofact__class_Ofact_000tc__Int__Oint,type,
    fact_fact_int: fun_int_int ).

tff(sy_c_Fact_Ofact__class_Ofact_000tc__Nat__Onat,type,
    fact_fact_nat: fun_nat_nat ).

tff(sy_c_Fields_Oinverse__class_Odivide_000tc__Complex__Ocomplex,type,
    invers1025623611omplex: fun_co822197010omplex ).

tff(sy_c_Fields_Oinverse__class_Odivide_000tc__Rat__Orat,type,
    inverse_divide_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Fields_Oinverse__class_Odivide_000tc__RealDef__Oreal,type,
    inverse_divide_real: fun_re229243589l_real ).

tff(sy_c_Fields_Oinverse__class_Oinverse_000tc__Complex__Ocomplex,type,
    invers1449016382omplex: fun_complex_complex ).

tff(sy_c_Fields_Oinverse__class_Oinverse_000tc__Rat__Orat,type,
    inverse_inverse_rat: fun_rat_rat ).

tff(sy_c_Fields_Oinverse__class_Oinverse_000tc__RealDef__Oreal,type,
    inverse_inverse_real: fun_real_real ).

tff(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
    finite_card_int: fun_fun_int_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
    finite_card_nat: fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1876863882t_bool: fun_fu236049833ol_nat ).

tff(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
    finite_finite_int: fun_fu819804446l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
    finite_finite_nat: fun_fu425979586l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1395289673t_bool: fun_fu1540142484l_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__Code____Numeral__Ocode____numeral_000tc__Code____Num,type,
    in_rel1251239042umeral: fun_Pr854501993l_bool > fun_Co1372573603l_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__Int__Oint_000tc__Int__Oint,type,
    in_rel_int_int: fun_Pr974702441t_bool > fun_int_fun_int_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,type,
    in_rel1239871883nt_int: fun_Pr1503267378t_bool > fun_in1698189120t_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__Nat__Onat_000tc__Nat__Onat,type,
    in_rel_nat_nat: fun_Pr1927655017t_bool > fun_nat_fun_nat_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Quickch,type,
    in_rel1707483778de_int: fun_Pr610342249t_bool > fun_Qu1069098747t_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    in_rel_real_real: fun_Pr1127657839l_bool > fun_re413263731l_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,type,
    in_rel512175208t_bool: fun_Pr4007113l_bool > fun_fu1151772339l_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc,type,
    in_rel682537726nt_int: fun_Pr1333588703t_bool > fun_Pr313471731t_bool ).

tff(sy_c_FunDef_Oin__rel_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_Itc,type,
    in_rel293241086at_nat: fun_Pr587612383t_bool > fun_Pr465597171t_bool ).

tff(sy_c_FunDef_Opair__leq,type,
    pair_leq: fun_Pr587612383t_bool ).

tff(sy_c_FunDef_Opair__less,type,
    pair_less: fun_Pr587612383t_bool ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod_int_int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_000tc__Int__Oint,type,
    gcd_gcd_int: fun_int_fun_int_int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_000tc__Nat__Onat,type,
    gcd_gcd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
    abs_abs_int: fun_int_int ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__Rat__Orat,type,
    abs_abs_rat: fun_rat_rat ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
    abs_abs_real: fun_real_real ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Code____Numeral__Ocode____numeral,type,
    minus_1690775515umeral: fun_Co1323339824umeral ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Complex__Ocomplex,type,
    minus_minus_complex: fun_co822197010omplex ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
    minus_minus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Quickcheck____Narrowing__Ocode____int,type,
    minus_534354567de_int: fun_Qu64793220de_int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Rat__Orat,type,
    minus_minus_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
    minus_minus_real: fun_re229243589l_real ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Code____Numeral__Ocode____numeral,type,
    one_on1645066479umeral: code_code_numeral ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Complex__Ocomplex,type,
    one_one_complex: complex ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
    one_one_int: int ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Quickcheck____Narrowing__Ocode____int,type,
    one_on1684967323de_int: quickcheck_code_int ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Rat__Orat,type,
    one_one_rat: rat ).

tff(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
    one_one_real: real ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Code____Numeral__Ocode____numeral,type,
    plus_p1627245867umeral: fun_Co1323339824umeral ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Complex__Ocomplex,type,
    plus_plus_complex: fun_co822197010omplex ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
    plus_plus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Quickcheck____Narrowing__Ocode____int,type,
    plus_p1446045655de_int: fun_Qu64793220de_int ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Rat__Orat,type,
    plus_plus_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
    plus_plus_real: fun_re229243589l_real ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__Complex__Ocomplex,type,
    sgn_sgn_complex: fun_complex_complex ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__Int__Oint,type,
    sgn_sgn_int: fun_int_int ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__Rat__Orat,type,
    sgn_sgn_rat: fun_rat_rat ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__RealDef__Oreal,type,
    sgn_sgn_real: fun_real_real ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Code____Numeral__Ocode____numeral,type,
    times_1655362735umeral: fun_Co1323339824umeral ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Complex__Ocomplex,type,
    times_times_complex: fun_co822197010omplex ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
    times_times_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Quickcheck____Narrowing__Ocode____int,type,
    times_123202395de_int: fun_Qu64793220de_int ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Rat__Orat,type,
    times_times_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
    times_times_real: fun_re229243589l_real ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Complex__Ocomplex,type,
    uminus473333897omplex: fun_complex_complex ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Int__Oint,type,
    uminus_uminus_int: fun_int_int ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Rat__Orat,type,
    uminus_uminus_rat: fun_rat_rat ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__RealDef__Oreal,type,
    uminus_uminus_real: fun_real_real ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    uminus407446779t_bool: fun_fu554825757t_bool ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    uminus1024588959t_bool: fun_fu821463397t_bool ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Code____Numeral__Ocode____numeral,type,
    zero_z126310315umeral: code_code_numeral ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Complex__Ocomplex,type,
    zero_zero_complex: complex ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
    zero_zero_int: int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
    zero_zero_nat: nat ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Quickcheck____Narrowing__Ocode____int,type,
    zero_z891286103de_int: quickcheck_code_int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Rat__Orat,type,
    zero_zero_rat: rat ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
    zero_zero_real: real ).

tff(sy_c_HOL_OThe_000tc__Int__Oint,type,
    the_int: fun_int_bool > int ).

tff(sy_c_HOL_OThe_000tc__RealDef__Oreal,type,
    the_real: fun_real_bool > real ).

tff(sy_c_HOL_OThe_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    the_Pr2103884470nt_int: fun_Pr974702441t_bool > product_prod_int_int ).

tff(sy_c_HOL_OThe_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    the_Pr588456374at_nat: fun_Pr1927655017t_bool > product_prod_nat_nat ).

tff(sy_c_Hilbert__Choice_OEps_000tc__Int__Oint,type,
    hilbert_Eps_int: fun_int_bool > int ).

tff(sy_c_Hilbert__Choice_OEps_000tc__RealDef__Oreal,type,
    hilbert_Eps_real: fun_real_bool > real ).

tff(sy_c_If_000tc__Int__Oint,type,
    if_int: fun_bo1947501544nt_int ).

tff(sy_c_If_000tc__Nat__Onat,type,
    if_nat: fun_bo1071572548at_nat ).

tff(sy_c_If_000tc__RealDef__Oreal,type,
    if_real: fun_bo1998411783l_real ).

tff(sy_c_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    if_Pro1731782967nt_int: fun_bo1552468991nt_int ).

tff(sy_c_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
    if_Pro313124157l_real: fun_bo1189210245l_real ).

tff(sy_c_Int2_OMultInv,type,
    multInv: int > fun_int_int ).

tff(sy_c_IntFact_Od22set,type,
    d22set: fun_int_fun_int_bool ).

tff(sy_c_IntFact_Ozfact,type,
    zfact: fun_int_int ).

tff(sy_c_IntPrimes_Oxzgcd,type,
    xzgcd: ( int * int ) > produc393999548nt_int ).

tff(sy_c_IntPrimes_Oxzgcda,type,
    xzgcda: ( int * int * int * int * int * int * int * int ) > produc393999548nt_int ).

tff(sy_c_IntPrimes_Ozcong,type,
    zcong: int > fun_int_fun_int_bool ).

tff(sy_c_IntPrimes_Ozprime,type,
    zprime: fun_int_bool ).

tff(sy_c_Int_OBit0,type,
    bit0: fun_int_int ).

tff(sy_c_Int_OBit1,type,
    bit1: fun_int_int ).

tff(sy_c_Int_OMin,type,
    min: int ).

tff(sy_c_Int_OPls,type,
    pls: int ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > fun_Pr974702441t_bool ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > fun_Pr974702441t_bool ).

tff(sy_c_Int_Oiszero_000tc__Int__Oint,type,
    iszero_int: fun_int_bool ).

tff(sy_c_Int_Oiszero_000tc__Rat__Orat,type,
    iszero_rat: fun_rat_bool ).

tff(sy_c_Int_Onat,type,
    nat_1: fun_int_nat ).

tff(sy_c_Int_Onat__aux,type,
    nat_aux: int > fun_nat_nat ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Code____Numeral__Ocode____numeral,type,
    number1443263063umeral: fun_in1022115617umeral ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Complex__Ocomplex,type,
    number528085621omplex: fun_int_complex ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
    number_number_of_int: fun_int_int ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
    number_number_of_nat: fun_int_nat ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Quickcheck____Narrowing__Ocode____int,type,
    number1226105091de_int: fun_in1505040589de_int ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Rat__Orat,type,
    number_number_of_rat: fun_int_rat ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
    number267125858f_real: fun_int_real ).

tff(sy_c_Int_Opred,type,
    pred: fun_int_int ).

tff(sy_c_Int_Oring__1__class_OInts_000tc__Complex__Ocomplex,type,
    ring_1_Ints_complex: fun_complex_bool ).

tff(sy_c_Int_Oring__1__class_OInts_000tc__Int__Oint,type,
    ring_1_Ints_int: fun_int_bool ).

tff(sy_c_Int_Oring__1__class_OInts_000tc__Rat__Orat,type,
    ring_1_Ints_rat: fun_rat_bool ).

tff(sy_c_Int_Oring__1__class_OInts_000tc__RealDef__Oreal,type,
    ring_1_Ints_real: fun_real_bool ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__Complex__Ocomplex,type,
    ring_11397209091omplex: fun_int_complex ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__Int__Oint,type,
    ring_1_of_int_int: fun_int_int ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__Rat__Orat,type,
    ring_1_of_int_rat: fun_int_rat ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__RealDef__Oreal,type,
    ring_1_of_int_real: fun_int_real ).

tff(sy_c_Int_Osucc,type,
    succ: fun_int_int ).

tff(sy_c_Lazy__Sequence_Osmall__lazy_H__rel,type,
    lazy_small_lazy_rel: fun_Pr313471731t_bool ).

tff(sy_c_Legacy__GCD_Ozgcd,type,
    legacy_zgcd: int > fun_int_int ).

tff(sy_c_Lim_OisCont_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
    isCont156215680omplex: fun_complex_complex > fun_complex_bool ).

tff(sy_c_Lim_OisCont_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
    isCont_complex_real: fun_complex_real > fun_complex_bool ).

tff(sy_c_Lim_OisCont_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    isCont_real_real: fun_real_real > fun_real_bool ).

tff(sy_c_Limits_Oat_000tc__Complex__Ocomplex,type,
    at_complex: complex > filter_complex ).

tff(sy_c_Limits_Oat_000tc__RealDef__Oreal,type,
    at_real: real > filter_real ).

tff(sy_c_Limits_Osequentially,type,
    sequentially: filter_nat ).

tff(sy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
    tendst1507391555omplex: ( fun_complex_complex * complex * filter_complex ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
    tendsto_complex_real: ( fun_complex_real * real * filter_complex ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex,type,
    tendsto_nat_complex: ( fun_nat_complex * complex * filter_nat ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal,type,
    tendsto_nat_real: ( fun_nat_real * real * filter_nat ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    tendsto_real_real: ( fun_real_real * real * filter_real ) > bool ).

tff(sy_c_Limits_Otrivial__limit_000tc__Nat__Onat,type,
    trivial_limit_nat: filter_nat > bool ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr313471731t_bool ).

tff(sy_c_Log_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Log_Opowr,type,
    powr: fun_re229243589l_real ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Onat__case_000tc__HOL__Obool,type,
    nat_case_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
    nat_case_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Onat__size,type,
    nat_size: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Complex__Ocomplex,type,
    semiri2020571505omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
    semiri1621563631at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Rat__Orat,type,
    semiri151668891at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__RealDef__Oreal,type,
    semiri132038758t_real: fun_nat_real ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Code____Numeral__Ocode____numeral,type,
    size_s945831648umeral: fun_Co116562173al_nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Complex__Ocomplex,type,
    size_size_complex: fun_complex_nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Int__Oint_J,type,
    size_size_list_int: fun_list_int_nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Nat__Onat,type,
    size_size_nat: fun_nat_nat ).

tff(sy_c_Nat__Numeral_Oneg,type,
    nat_neg: fun_int_bool ).

tff(sy_c_Nat__Transfer_Ois__nat,type,
    nat_is_nat: fun_int_bool ).

tff(sy_c_Nat__Transfer_Onat__set,type,
    nat_nat_set: fun_fu819804446l_bool ).

tff(sy_c_Nat__Transfer_Otransfer__morphism_000tc__Int__Oint_000tc__Nat__Onat,type,
    nat_tr876908586nt_nat: fun_int_nat > fun_fu819804446l_bool ).

tff(sy_c_Nat__Transfer_Otransfer__morphism_000tc__Nat__Onat_000tc__Int__Oint,type,
    nat_tr160667106at_int: fun_nat_int > fun_fu425979586l_bool ).

tff(sy_c_Nat__Transfer_Otsub,type,
    nat_tsub: int > fun_int_int ).

tff(sy_c_Nitpick_OFrac,type,
    frac: fun_Pr974702441t_bool ).

tff(sy_c_Nitpick_Oint__gcd,type,
    int_gcd: fun_int_fun_int_int ).

tff(sy_c_Nitpick_Oint__lcm,type,
    int_lcm: int > fun_int_int ).

tff(sy_c_Nitpick_Onat__gcd,type,
    nat_gcd: nat > fun_nat_nat ).

tff(sy_c_Nitpick_Onat__gcd__rel,type,
    nat_gcd_rel: fun_Pr465597171t_bool ).

tff(sy_c_Nitpick_Onat__lcm,type,
    nat_lcm: nat > fun_nat_nat ).

tff(sy_c_Nitpick_Onorm__frac,type,
    norm_frac: ( int * int ) > product_prod_int_int ).

tff(sy_c_Nitpick_Onorm__frac__rel,type,
    norm_frac_rel: fun_Pr313471731t_bool ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na182379510l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Code____Numeral__Ocode____numeral,type,
    ord_le1304079648umeral: fun_Co1372573603l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Quickcheck____Narrowing__Ocode____int,type,
    ord_le1860547276de_int: fun_Qu1069098747t_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
    ord_less_real: fun_re413263731l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le1912455174t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le382113706t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Code____Numeral__Ocode____numeral,type,
    ord_le565307924umeral: fun_Co1372573603l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__HOL__Obool,type,
    ord_less_eq_bool: fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Quickcheck____Narrowing__Ocode____in,type,
    ord_le258702272de_int: fun_Qu1069098747t_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
    ord_less_eq_real: fun_re413263731l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le951220754t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le1568362934t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Parity_Oeven__odd__class_Oeven_000tc__Int__Oint,type,
    even_odd_even_int: fun_int_bool ).

tff(sy_c_Parity_Oeven__odd__class_Oeven_000tc__Nat__Onat,type,
    even_odd_even_nat: fun_nat_bool ).

tff(sy_c_Power_Opower__class_Opower_000tc__Complex__Ocomplex,type,
    power_power_complex: fun_co777804144omplex ).

tff(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
    power_power_real: fun_re1927296998t_real ).

tff(sy_c_Primes_Ocoprime,type,
    coprime: fun_nat_fun_nat_bool ).

tff(sy_c_Primes_Ofact,type,
    fact: fun_nat_nat ).

tff(sy_c_Primes_Oprime,type,
    prime: fun_nat_bool ).

tff(sy_c_Product__Type_OPair_000tc__Code____Numeral__Ocode____numeral_000tc__Code___,type,
    produc2136830103umeral: fun_Co1101317613umeral ).

tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
    product_Pair_int_int: fun_in1111845413nt_int ).

tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__O,type,
    produc282740534nt_int: fun_in1560792517nt_int ).

tff(sy_c_Product__Type_OPair_000tc__Nat__Onat_000tc__Nat__Onat,type,
    product_Pair_nat_nat: fun_na257798253at_nat ).

tff(sy_c_Product__Type_OPair_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Qui,type,
    produc1318306967de_int: fun_Qu1353705541de_int ).

tff(sy_c_Product__Type_OPair_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc865579683l_real: fun_re1550493891l_real ).

tff(sy_c_Product__Type_OPair_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_,type,
    produc713279741t_bool: fun_fu497932381t_bool ).

tff(sy_c_Product__Type_OPair_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod,type,
    produc883642259nt_int: fun_Pr668393395nt_int ).

tff(sy_c_Product__Type_OPair_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod,type,
    produc494345619at_nat: fun_Pr1071803827at_nat ).

tff(sy_c_Product__Type_Oapsnd_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint,type,
    produc713050258nt_int: fun_int_int > fun_Pr241715507nt_int ).

tff(sy_c_Product__Type_Ocurry_000tc__Code____Numeral__Ocode____numeral_000tc__Code__,type,
    produc609372286l_bool: fun_Pr854501993l_bool > fun_Co1372573603l_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool,type,
    produc262399358t_bool: fun_Pr974702441t_bool > fun_int_fun_int_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__,type,
    produc97574839t_bool: fun_Pr1503267378t_bool > fun_in1698189120t_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool,type,
    produc2091540094t_bool: fun_Pr1927655017t_bool > fun_nat_fun_nat_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Qu,type,
    produc2013771390t_bool: fun_Pr610342249t_bool > fun_Qu1069098747t_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL,type,
    produc748992778l_bool: fun_Pr1127657839l_bool > fun_re413263731l_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun,type,
    produc1775888740l_bool: fun_Pr4007113l_bool > fun_fu1151772339l_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__pro,type,
    produc353512698t_bool: fun_Pr1333588703t_bool > fun_Pr313471731t_bool ).

tff(sy_c_Product__Type_Ocurry_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__pro,type,
    produc1319041786t_bool: fun_Pr587612383t_bool > fun_Pr465597171t_bool ).

tff(sy_c_Product__Type_Ofst_000tc__Code____Numeral__Ocode____numeral_000tc__Code____,type,
    produc310214954umeral: fun_Pr1092513364umeral ).

tff(sy_c_Product__Type_Ofst_000tc__Int__Oint_000tc__Int__Oint,type,
    product_fst_int_int: fun_Pr480351984nt_int ).

tff(sy_c_Product__Type_Ofst_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oi,type,
    produc2110120675nt_int: fun_Pr490082791nt_int ).

tff(sy_c_Product__Type_Ofst_000tc__Nat__Onat_000tc__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr1434736532at_nat ).

tff(sy_c_Product__Type_Ofst_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Quic,type,
    produc1968959274de_int: fun_Pr637471232de_int ).

tff(sy_c_Product__Type_Ofst_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc1935615926l_real: fun_Pr608224961l_real ).

tff(sy_c_Product__Type_Ofst_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_I,type,
    produc926076176t_bool: fun_Pr523861064t_bool ).

tff(sy_c_Product__Type_Ofst_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_,type,
    produc1096438694nt_int: fun_Pr851629481nt_int ).

tff(sy_c_Product__Type_Ofst_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_,type,
    produc707142054at_nat: fun_Pr639240873at_nat ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__HO,type,
    produc93441119t_bool: fun_int_fun_int_bool > fun_Pr974702441t_bool ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__In,type,
    produc1298267108nt_int: fun_int_fun_int_int > fun_Pr480351984nt_int ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__pr,type,
    produc1518849193nt_int: fun_in1111845413nt_int > fun_Pr241715507nt_int ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HO,type,
    produc1922581855t_bool: fun_nat_fun_nat_bool > fun_Pr1927655017t_bool ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Nat__Onat_000tc__Nat__Onat_000tc__pr,type,
    produc1391996073at_nat: fun_na257798253at_nat > fun_Pr426809139at_nat ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc595218619l_real: fun_re1550493891l_real > fun_Pr1375198143l_real ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    produc973645403t_bool: fun_Pr313471731t_bool > fun_Pr1333588703t_bool ).

tff(sy_c_Product__Type_Osnd_000tc__Code____Numeral__Ocode____numeral_000tc__Code____,type,
    produc1857364076umeral: fun_Pr1092513364umeral ).

tff(sy_c_Product__Type_Osnd_000tc__Int__Oint_000tc__Int__Oint,type,
    product_snd_int_int: fun_Pr480351984nt_int ).

tff(sy_c_Product__Type_Osnd_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oi,type,
    produc8341793nt_int: fun_Pr1086341500nt_int ).

tff(sy_c_Product__Type_Osnd_000tc__Nat__Onat_000tc__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr1434736532at_nat ).

tff(sy_c_Product__Type_Osnd_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Quic,type,
    produc221809260de_int: fun_Pr637471232de_int ).

tff(sy_c_Product__Type_Osnd_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc556554744l_real: fun_Pr608224961l_real ).

tff(sy_c_Product__Type_Osnd_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_I,type,
    produc840788306t_bool: fun_Pr523861064t_bool ).

tff(sy_c_Product__Type_Osnd_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_,type,
    produc1011150824nt_int: fun_Pr851629481nt_int ).

tff(sy_c_Product__Type_Osnd_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_,type,
    produc621854184at_nat: fun_Pr639240873at_nat ).

tff(sy_c_Quickcheck__Narrowing_Oaround__zero,type,
    quickc666637781d_zero: int > list_int ).

tff(sy_c_Quickcheck__Narrowing_Oaround__zero__rel,type,
    quickc1265749348ro_rel: fun_int_fun_int_bool ).

tff(sy_c_Quickcheck__Narrowing_Odiv__mod__code__int,type,
    quickc495462417de_int: ( quickcheck_code_int * quickcheck_code_int ) > produc167071911de_int ).

tff(sy_c_Quickcheck__Narrowing_Oint__of,type,
    quickcheck_int_of: quickcheck_code_int > int ).

tff(sy_c_Quickcheck__Narrowing_Onat__of,type,
    quickcheck_nat_of: quickcheck_code_int > nat ).

tff(sy_c_Quickcheck__Narrowing_Oof__int,type,
    quickcheck_of_int: fun_in1505040589de_int ).

tff(sy_c_RComplete_Onatceiling,type,
    natceiling: real > nat ).

tff(sy_c_RComplete_Onatfloor,type,
    natfloor: real > nat ).

tff(sy_c_Rat_OFract,type,
    fract: fun_int_fun_int_rat ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_000tc__RealDef__Oreal,type,
    field_1210416355s_real: fun_real_bool ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr1333588703t_bool ).

tff(sy_c_RealDef_ORatreal,type,
    ratreal: rat > real ).

tff(sy_c_RealDef_Oreal_000tc__Int__Oint,type,
    real_int: fun_int_real ).

tff(sy_c_RealDef_Oreal_000tc__Nat__Onat,type,
    real_nat: fun_nat_real ).

tff(sy_c_RealDef_Ovanishes,type,
    vanishes: fun_nat_rat > bool ).

tff(sy_c_RealVector_Odist__class_Odist_000tc__Complex__Ocomplex,type,
    dist_dist_complex: complex > fun_complex_real ).

tff(sy_c_RealVector_Odist__class_Odist_000tc__RealDef__Oreal,type,
    dist_dist_real: real > fun_real_real ).

tff(sy_c_RealVector_Onorm__class_Onorm_000tc__Complex__Ocomplex,type,
    norm_norm_complex: fun_complex_real ).

tff(sy_c_RealVector_Onorm__class_Onorm_000tc__RealDef__Oreal,type,
    norm_norm_real: fun_real_real ).

tff(sy_c_RealVector_Oof__real_000tc__Complex__Ocomplex,type,
    of_real_complex: real > complex ).

tff(sy_c_RealVector_OscaleR__class_OscaleR_000tc__Complex__Ocomplex,type,
    scaleR1652505878omplex: real > fun_complex_complex ).

tff(sy_c_RealVector_OscaleR__class_OscaleR_000tc__RealDef__Oreal,type,
    scaleR_scaleR_real: real > fun_real_real ).

tff(sy_c_Residues_OLegendre,type,
    legendre: int > fun_int_int ).

tff(sy_c_Residues_OQuadRes,type,
    quadRes: fun_int_fun_int_bool ).

tff(sy_c_Residues_OResSet,type,
    resSet: int > fun_fu819804446l_bool ).

tff(sy_c_Residues_OSR,type,
    sr: fun_int_fun_int_bool ).

tff(sy_c_Residues_OSRStar,type,
    sRStar: fun_int_fun_int_bool ).

tff(sy_c_Residues_OStandardRes,type,
    standardRes: int > fun_int_int ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_SEQ_OBseq_000tc__RealDef__Oreal,type,
    bseq_real: fun_nat_real > bool ).

tff(sy_c_SEQ_OCauchy_000tc__Complex__Ocomplex,type,
    cauchy_complex: fun_nat_complex > bool ).

tff(sy_c_SEQ_OCauchy_000tc__RealDef__Oreal,type,
    cauchy_real: fun_nat_real > bool ).

tff(sy_c_SEQ_Omonoseq_000tc__RealDef__Oreal,type,
    monoseq_real: fun_nat_real > bool ).

tff(sy_c_SMT_Oz3div,type,
    z3div: int > fun_int_int ).

tff(sy_c_SMT_Oz3mod,type,
    z3mod: int > fun_int_int ).

tff(sy_c_Series_Osuminf_000tc__Complex__Ocomplex,type,
    suminf_complex: fun_fu1930053706omplex ).

tff(sy_c_Series_Osuminf_000tc__RealDef__Oreal,type,
    suminf_real: fun_fu1122561126l_real ).

tff(sy_c_Series_Osummable_000tc__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > bool ).

tff(sy_c_Series_Osummable_000tc__RealDef__Oreal,type,
    summable_real: fun_nat_real > bool ).

tff(sy_c_Series_Osums_000tc__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_bool ).

tff(sy_c_Series_Osums_000tc__RealDef__Oreal,type,
    sums_real: fun_nat_real > fun_real_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Int__Oint,type,
    ord_at875362053st_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Nat__Onat,type,
    ord_at238088361st_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__RealDef__Oreal,type,
    ord_at1589558736t_real: real > fun_re413263731l_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Int__Oint,type,
    ord_at641636577an_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Nat__Onat,type,
    ord_at4362885an_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__RealDef__Oreal,type,
    ord_at1496968948n_real: real > fun_re413263731l_bool ).

tff(sy_c_SetInterval_Oord__class_OatMost_000tc__Nat__Onat,type,
    ord_atMost_nat: fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Int__Oint,type,
    ord_gr1297742076an_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Nat__Onat,type,
    ord_gr660468384an_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__RealDef__Oreal,type,
    ord_gr788844697n_real: real > fun_re413263731l_bool ).

tff(sy_c_SetInterval_Oord__class_OlessThan_000tc__Nat__Onat,type,
    ord_lessThan_nat: fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OlessThan_000tc__RealDef__Oreal,type,
    ord_lessThan_real: fun_re413263731l_bool ).

tff(sy_c_Set_OCollect_000tc__Int__Oint,type,
    collect_int: fun_int_bool > fun_int_bool ).

tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
    collect_nat: fun_nat_bool > fun_nat_bool ).

tff(sy_c_Set_OCollect_000tc__RealDef__Oreal,type,
    collect_real: fun_real_bool > fun_real_bool ).

tff(sy_c_Set_OCollect_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    collec1347809874nt_int: fun_Pr974702441t_bool > fun_Pr974702441t_bool ).

tff(sy_c_Set_OCollect_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    collec1979865426at_nat: fun_Pr1927655017t_bool > fun_Pr1927655017t_bool ).

tff(sy_c_Set_OCollect_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pro,type,
    collec50511176nt_int: fun_Pr1333588703t_bool > fun_Pr1333588703t_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Int__Oint,type,
    image_int_int: ( fun_int_int * fun_int_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * fun_int_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    image_275383677t_bool: ( fun_int_fun_int_bool * fun_int_bool ) > fun_fu819804446l_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
    image_nat_int: ( fun_nat_int * fun_nat_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Ocos,type,
    cos: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Odiffs_000tc__RealDef__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_000tc__RealDef__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln,type,
    ln: fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Osin,type,
    sin: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Otan,type,
    tan: fun_real_real ).

tff(sy_c_TwoSquares__Mirabelle__tyfmylkaxk_Ois__sum2sq,type,
    twoSqu1425020248sum2sq: fun_int_bool ).

tff(sy_c_TwoSquares__Mirabelle__tyfmylkaxk_Osum2sq,type,
    twoSqu2000912462sum2sq: fun_Pr480351984nt_int ).

tff(sy_c_Wellfounded_Oaccp_000tc__Int__Oint,type,
    accp_int: fun_int_fun_int_bool > fun_int_bool ).

tff(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    accp_P2006205492nt_int: fun_Pr313471731t_bool > fun_Pr974702441t_bool ).

tff(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    accp_P490777396at_nat: fun_Pr465597171t_bool > fun_Pr1927655017t_bool ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: fun_Pr1927655017t_bool ).

tff(sy_c_WilsonRuss_Oinv,type,
    inv: int > fun_int_int ).

tff(sy_c_WilsonRuss_Owset,type,
    wset: int > fun_int_fun_int_bool ).

tff(sy_c_fEx_000tc__Int__Oint,type,
    fEx_int: fun_fu819804446l_bool ).

tff(sy_c_fEx_000tc__Nat__Onat,type,
    fEx_nat: fun_fu425979586l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: fun_bo1549164019l_bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal_000tc__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_000tc__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_000tc__Rat__Orat,type,
    fequal_rat: fun_rat_fun_rat_bool ).

tff(sy_c_fequal_000tc__RealDef__Oreal,type,
    fequal_real: fun_re413263731l_bool ).

tff(sy_c_fimplies,type,
    fimplies: bool > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__Code____Numeral__Ocode,type,
    hAPP_C498520661umeral: ( fun_Co536835517umeral * code_code_numeral ) > code_code_numeral ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__HOL__Obool,type,
    hAPP_C24501650l_bool: ( fun_Co1584449344l_bool * code_code_numeral ) > bool ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__Nat__Onat,type,
    hAPP_C2129356693al_nat: ( fun_Co116562173al_nat * code_code_numeral ) > nat ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Code____Numer,type,
    hAPP_C1594335432umeral: ( fun_Co1323339824umeral * code_code_numeral ) > fun_Co536835517umeral ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Code____Numer_262,type,
    hAPP_C10902773l_bool: ( fun_Co1372573603l_bool * code_code_numeral ) > fun_Co1584449344l_bool ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Code____Numer_263,type,
    hAPP_C1614127039umeral: ( fun_Co1101317613umeral * code_code_numeral ) > fun_Co991465098umeral ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__prod_Itc__Code____Nume,type,
    hAPP_C2132160860umeral: ( fun_Co991465098umeral * code_code_numeral ) > produc1359518119umeral ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
    hAPP_complex_complex: ( fun_complex_complex * complex ) > complex ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__HOL__Obool,type,
    hAPP_complex_bool: ( fun_complex_bool * complex ) > bool ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__Nat__Onat,type,
    hAPP_complex_nat: ( fun_complex_nat * complex ) > nat ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
    hAPP_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__fun_Itc__Complex__Ocomplex_Mtc__Comple,type,
    hAPP_c172932010omplex: ( fun_co822197010omplex * complex ) > fun_complex_complex ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__fun_Itc__Nat__Onat_Mtc__Complex__Ocomp,type,
    hAPP_c1088319240omplex: ( fun_co777804144omplex * complex ) > fun_nat_complex ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__fun_Itc__fun_Itc__Complex__Ocomplex_Mt,type,
    hAPP_c769186317l_bool: ( fun_co1518175989l_bool * complex ) > fun_fu672930208l_bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
    hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mt,type,
    hAPP_b1463609396nt_int: ( fun_bo1947501544nt_int * bool ) > fun_int_fun_int_int ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mt,type,
    hAPP_b992065680at_nat: ( fun_bo1071572548at_nat * bool ) > fun_nat_fun_nat_nat ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef,type,
    hAPP_b646336293l_real: ( fun_bo1998411783l_real * bool ) > fun_re229243589l_real ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    hAPP_b40753821nt_int: ( fun_bo1552468991nt_int * bool ) > fun_Pr1580971581nt_int ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDe,type,
    hAPP_b1482221219l_real: ( fun_bo1189210245l_real * bool ) > fun_Pr1283822147l_real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Code____Numeral__Ocode____numeral,type,
    hAPP_i769753017umeral: ( fun_in1022115617umeral * int ) > code_code_numeral ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Complex__Ocomplex,type,
    hAPP_int_complex: ( fun_int_complex * int ) > complex ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
    hAPP_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,type,
    hAPP_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Nat__Onat,type,
    hAPP_int_nat: ( fun_int_nat * int ) > nat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Quickcheck____Narrowing__Ocode____int,type,
    hAPP_i1732201573de_int: ( fun_in1505040589de_int * int ) > quickcheck_code_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Rat__Orat,type,
    hAPP_int_rat: ( fun_int_rat * int ) > rat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__RealDef__Oreal,type,
    hAPP_int_real: ( fun_int_real * int ) > real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_i68813070l_bool: ( fun_in531499254l_bool * int ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    hAPP_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J,type,
    hAPP_int_fun_int_rat: ( fun_int_fun_int_rat * int ) > fun_int_rat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J,type,
    hAPP_i627165055t_real: ( fun_int_fun_int_real * int ) > fun_int_real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mt,type,
    hAPP_i1823918693l_bool: ( fun_in397122125l_bool * int ) > fun_in531499254l_bool ).

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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_418,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_419,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_420,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__Nat__Onat_J_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__Int,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__Int_421,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int__422,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int__423,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__O,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__O_425,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__O_426,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_It,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod_I,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__O_427,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_Mt,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_Mt,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int__Oint_M,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__RealDef__Ore,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__Nat,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__429,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__430,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__431,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_Mtc__fun_Itc,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_Mtc__fun_Itc_432,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_Mtc__fun_Itc_433,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_434,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_435,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_436,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_M,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__RealDe,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__RealDe_438,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mtc_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J,type,
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    hAPP_f125837377t_bool: ( fun_fu620454517t_bool * fun_Pr1254492423t_bool ) > fun_fu1859693515t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_455,type,
    hAPP_f11709033t_bool: ( fun_fu2062813463t_bool * fun_Pr755126262t_bool ) > fun_fu104040328t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_456,type,
    hAPP_f1769710159nt_int: ( fun_fu1223580849nt_int * fun_Pr642042703nt_int ) > fun_fu1123997949nt_int ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro,type,
    hAPP_f1781059817t_bool: ( fun_fu1859693515t_bool * fun_Pr313471731t_bool ) > fun_Pr313471731t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro_457,type,
    hAPP_f1187443546t_bool: ( fun_fu104040328t_bool * fun_Pr2144253564nt_int ) > fun_Pr313471731t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro_458,type,
    hAPP_f1198736473t_bool: ( fun_fu550659463t_bool * fun_Pr2144253564nt_int ) > fun_Pr820506994t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro_459,type,
    hAPP_f223623489t_bool: ( fun_fu694788213t_bool * fun_Pr820506994t_bool ) > fun_Pr755126262t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__In,type,
    hAPP_f151095827nt_int: ( fun_fu2102390523nt_int * fun_Pr241715507nt_int ) > fun_fu1086065329nt_int ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int_,type,
    hAPP_f928428050l_bool: ( fun_fu52199540l_bool * fun_Pr1503267378t_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__HOL__Obool_J,type,
    hAPP_f879361201l_bool: ( fun_fu56396127l_bool * fun_Pr1927655017t_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Q,type,
    hAPP_f222989233l_bool: ( fun_fu1693254751l_bool * fun_Pr610342249t_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__HO,type,
    hAPP_f46817975l_bool: ( fun_fu35639397l_bool * fun_Pr1127657839l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__HO_460,type,
    hAPP_f940061907l_bool: ( fun_fu223014715l_bool * fun_Pr1127657839l_bool ) > fun_fu1280874925l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__Re,type,
    hAPP_f918606171t_real: ( fun_fu1983483971t_real * fun_Pr608224961l_real ) > fun_fu2095711751t_real ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fu,type,
    hAPP_f1469637905l_bool: ( fun_fu2145628351l_bool * fun_Pr4007113l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pr,type,
    hAPP_f526364711l_bool: ( fun_fu1202355157l_bool * fun_Pr1333588703t_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__pr,type,
    hAPP_f115998247l_bool: ( fun_fu791988693l_bool * fun_Pr587612383t_bool ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Numera,type,
    hAPP_P619277728umeral: ( fun_Pr1092513364umeral * produc1359518119umeral ) > code_code_numeral ).

tff(sy_c_hAPP_000tc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Numera_461,type,
    hAPP_P1164677895l_bool: ( fun_Pr854501993l_bool * produc1359518119umeral ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Numera_462,type,
    hAPP_P2045320969l_bool: ( fun_Pr1402959211l_bool * produc1359518119umeral ) > fun_fu1989823839l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,type,
    hAPP_P603027463t_bool: ( fun_Pr974702441t_bool * product_prod_int_int ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint,type,
    hAPP_P1175774780nt_int: ( fun_Pr480351984nt_int * product_prod_int_int ) > int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Rat__Orat,type,
    hAPP_P1853363688nt_rat: ( fun_Pr1024438428nt_rat * product_prod_int_int ) > rat ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__HOL__Obool,type,
    hAPP_P921862901l_bool: ( fun_Pr189909033l_bool * product_prod_int_int ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint_,type,
    hAPP_P252494982t_bool: ( fun_Pr1149636328t_bool * product_prod_int_int ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint__463,type,
    hAPP_P2110489235nt_int: ( fun_Pr2134412615nt_int * product_prod_int_int ) > fun_int_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint__464,type,
    hAPP_P507092031nt_rat: ( fun_Pr1826394355nt_rat * product_prod_int_int ) > fun_int_rat ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint__465,type,
    hAPP_P621635040nt_int: ( fun_Pr1238001172nt_int * product_prod_int_int ) > fun_in989261322nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p,type,
    hAPP_P1268961545l_bool: ( fun_Pr1229452139l_bool * product_prod_int_int ) > fun_fu28254815l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p_466,type,
    hAPP_P877703507t_bool: ( fun_Pr1254492423t_bool * product_prod_int_int ) > fun_fu2112443261t_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p_467,type,
    hAPP_P1400155668t_bool: ( fun_Pr755126262t_bool * product_prod_int_int ) > fun_fu1337909610t_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p_468,type,
    hAPP_P752050971nt_int: ( fun_Pr642042703nt_int * product_prod_int_int ) > fun_fu1494809157nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc__,type,
    hAPP_P989584703t_bool: ( fun_Pr313471731t_bool * product_prod_int_int ) > fun_Pr974702441t_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___469,type,
    hAPP_P1668131738nt_int: ( fun_Pr2144253564nt_int * product_prod_int_int ) > fun_Pr480351984nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___470,type,
    hAPP_P2053772734t_bool: ( fun_Pr820506994t_bool * product_prod_int_int ) > fun_Pr1149636328t_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___471,type,
    hAPP_P1145851913nt_int: ( fun_Pr1580971581nt_int * product_prod_int_int ) > fun_Pr241715507nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___472,type,
    hAPP_P941956607nt_int: ( fun_Pr668393395nt_int * product_prod_int_int ) > fun_Pr29212713nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc__Int__Oint,type,
    hAPP_P1975530577nt_int: ( fun_Pr241715507nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc__Int__Oint_473,type,
    hAPP_P408881810nt_int: ( fun_Pr1076553414nt_int * product_prod_int_int ) > produc393999548nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc__prod_Itc_,type,
    hAPP_P230595783nt_int: ( fun_Pr29212713nt_int * product_prod_int_int ) > produc1137372701nt_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
    hAPP_P178748804t_bool: ( fun_Pr1503267378t_bool * produc393999548nt_int ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J_474,type,
    hAPP_P1251258751nt_int: ( fun_Pr490082791nt_int * produc393999548nt_int ) > int ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J_475,type,
    hAPP_P2074576781l_bool: ( fun_Pr750055925l_bool * produc393999548nt_int ) > fun_fu52199540l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J_476,type,
    hAPP_P779853134nt_int: ( fun_Pr1086341500nt_int * produc393999548nt_int ) > product_prod_int_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__HOL__Obool,type,
    hAPP_P1555980039t_bool: ( fun_Pr1927655017t_bool * product_prod_nat_nat ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__Nat__Onat,type,
    hAPP_P1654798560at_nat: ( fun_Pr1434736532at_nat * product_prod_nat_nat ) > nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc__fun_Itc__p,type,
    hAPP_P1251024649l_bool: ( fun_Pr2045952363l_bool * product_prod_nat_nat ) > fun_fu56396127l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc__prod_Itc__,type,
    hAPP_P705731903t_bool: ( fun_Pr465597171t_bool * product_prod_nat_nat ) > fun_Pr1927655017t_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc__prod_Itc___477,type,
    hAPP_P402336767at_nat: ( fun_Pr1071803827at_nat * product_prod_nat_nat ) > fun_Pr1858073897at_nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_Itc__Nat__Onat,type,
    hAPP_P1586233937at_nat: ( fun_Pr426809139at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_Itc__prod_Itc_,type,
    hAPP_P369611207at_nat: ( fun_Pr1858073897at_nat * product_prod_nat_nat ) > produc1322466333at_nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickcheck,type,
    hAPP_P1709355527t_bool: ( fun_Pr610342249t_bool * produc167071911de_int ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickcheck_478,type,
    hAPP_P1913525324de_int: ( fun_Pr637471232de_int * produc167071911de_int ) > quickcheck_code_int ).

tff(sy_c_hAPP_000tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickcheck_479,type,
    hAPP_P1879777545l_bool: ( fun_Pr1589974379l_bool * produc167071911de_int ) > fun_fu1693254751l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__HOL__Oboo,type,
    hAPP_P1333854989l_bool: ( fun_Pr1127657839l_bool * produc914805421l_real ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__RealDef__,type,
    hAPP_P731461727l_real: ( fun_Pr608224961l_real * produc914805421l_real ) > real ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__fun_Itc__,type,
    hAPP_P1946311189l_bool: ( fun_Pr1405411063l_bool * produc914805421l_real ) > fun_fu35639397l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__fun_Itc___480,type,
    hAPP_P1982799375l_real: ( fun_Pr1283822147l_real * produc914805421l_real ) > fun_Pr1375198143l_real ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__prod_Itc_,type,
    hAPP_P1860904029l_real: ( fun_Pr1375198143l_real * produc914805421l_real ) > produc914805421l_real ).

tff(sy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__In,type,
    hAPP_P880235623l_bool: ( fun_Pr4007113l_bool * produc1450482951t_bool ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__In_481,type,
    hAPP_P1477980390t_bool: ( fun_Pr523861064t_bool * produc1450482951t_bool ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__In_482,type,
    hAPP_P1681069257l_bool: ( fun_Pr1669804459l_bool * produc1450482951t_bool ) > fun_fu2145628351l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__I,type,
    hAPP_P62333565t_bool: ( fun_Pr1333588703t_bool * produc1137372701nt_int ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__I_483,type,
    hAPP_P1062890741l_bool: ( fun_Pr1020157655l_bool * produc1137372701nt_int ) > fun_fu1202355157l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__I_484,type,
    hAPP_P141737927nt_int: ( fun_Pr851629481nt_int * produc1137372701nt_int ) > product_prod_int_int ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__N,type,
    hAPP_P1463840893t_bool: ( fun_Pr587612383t_bool * produc1322466333at_nat ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__N_485,type,
    hAPP_P1876494581l_bool: ( fun_Pr822475479l_bool * produc1322466333at_nat ) > fun_fu791988693l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__N_486,type,
    hAPP_P1502442695at_nat: ( fun_Pr639240873at_nat * produc1322466333at_nat ) > product_prod_nat_nat ).

tff(sy_c_hBOOL,type,
    hBOOL: bool > $o ).

tff(sy_c_member_000tc__Complex__Ocomplex,type,
    member_complex: fun_co1518175989l_bool ).

tff(sy_c_member_000tc__Int__Oint,type,
    member_int: fun_in938817141l_bool ).

tff(sy_c_member_000tc__Nat__Onat,type,
    member_nat: fun_na1436237685l_bool ).

tff(sy_c_member_000tc__Rat__Orat,type,
    member_rat: fun_ra1984299381l_bool ).

tff(sy_c_member_000tc__RealDef__Oreal,type,
    member_real: fun_re1865242107l_bool ).

tff(sy_c_member_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    member_fun_int_bool: fun_fu1637550549l_bool ).

tff(sy_c_member_000tc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Nume,type,
    member1246959636umeral: fun_Pr1402959211l_bool ).

tff(sy_c_member_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    member1747683604nt_int: fun_Pr1229452139l_bool ).

tff(sy_c_member_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    member791087631nt_int: fun_Pr750055925l_bool ).

tff(sy_c_member_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    member232255508at_nat: fun_Pr2045952363l_bool ).

tff(sy_c_member_000tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickche,type,
    member2092545300de_int: fun_Pr1589974379l_bool ).

tff(sy_c_member_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
    member1344826522l_real: fun_Pr1405411063l_bool ).

tff(sy_c_member_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__,type,
    member308914164t_bool: fun_Pr1669804459l_bool ).

tff(sy_c_member_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc_,type,
    member2143287562nt_int: fun_Pr1020157655l_bool ).

tff(sy_c_member_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc_,type,
    member180897546at_nat: fun_Pr822475479l_bool ).

tff(sy_v_a,type,
    a: int ).

tff(sy_v_b,type,
    b: int ).

tff(sy_v_p,type,
    p: int ).

tff(sy_v_q,type,
    q: int ).

%----Relevant facts (5196)
tff(fact_0_xzgcda__linear__aux1,axiom,
    ! [A_2: int,R_3: int,B_1: int,M_1: int,C_1: int,D_3: int,N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),B_1))),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),D_3))),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),N_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_3),N_1)))) ).

tff(fact_1_mult__diff__mult,axiom,
    ! [X_4: rat,Y_4: rat,A_12: rat,B_10: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_4),Y_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_12),B_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_4),B_10))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_4),A_12)),B_10)) ).

tff(fact_2_mult__diff__mult,axiom,
    ! [X_4: complex,Y_4: complex,A_12: complex,B_10: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_4),Y_4)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_12),B_10)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_4),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_4),B_10))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_4),A_12)),B_10)) ).

tff(fact_3_mult__diff__mult,axiom,
    ! [X_4: real,Y_4: real,A_12: real,B_10: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_4),Y_4)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_12),B_10)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_4),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_4),B_10))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_4),A_12)),B_10)) ).

tff(fact_4_mult__diff__mult,axiom,
    ! [X_4: int,Y_4: int,A_12: int,B_10: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_4),Y_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_12),B_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_4),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_4),B_10))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_4),A_12)),B_10)) ).

tff(fact_5_mult_Oprod__diff__prod,axiom,
    ! [X_1: complex,Y_2: complex,A_249: complex,B_194: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_249),B_194)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),A_249)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_2),B_194))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),A_249)),B_194))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_249),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_2),B_194))) ).

tff(fact_6_mult_Oprod__diff__prod,axiom,
    ! [X_1: real,Y_2: real,A_249: real,B_194: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_249),B_194)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),A_249)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),B_194))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),A_249)),B_194))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_249),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),B_194))) ).

tff(fact_7_eq__add__iff2,axiom,
    ! [Aa: rat,E_2: rat,C: rat,Ba: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),E_2)),D) )
    <=> ( C = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Ba),Aa)),E_2)),D) ) ) ).

tff(fact_8_eq__add__iff2,axiom,
    ! [Aa: complex,E_2: complex,C: complex,Ba: complex,D: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ba),E_2)),D) )
    <=> ( C = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Ba),Aa)),E_2)),D) ) ) ).

tff(fact_9_eq__add__iff2,axiom,
    ! [Aa: real,E_2: real,C: real,Ba: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),E_2)),D) )
    <=> ( C = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)),E_2)),D) ) ) ).

tff(fact_10_eq__add__iff2,axiom,
    ! [Aa: int,E_2: int,C: int,Ba: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E_2)),D) )
    <=> ( C = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),Aa)),E_2)),D) ) ) ).

tff(fact_11_eq__add__iff1,axiom,
    ! [Aa: rat,E_2: rat,C: rat,Ba: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),E_2)),D) )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba)),E_2)),C) = D ) ) ).

tff(fact_12_eq__add__iff1,axiom,
    ! [Aa: complex,E_2: complex,C: complex,Ba: complex,D: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ba),E_2)),D) )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Aa),Ba)),E_2)),C) = D ) ) ).

tff(fact_13_eq__add__iff1,axiom,
    ! [Aa: real,E_2: real,C: real,Ba: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),E_2)),D) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba)),E_2)),C) = D ) ) ).

tff(fact_14_eq__add__iff1,axiom,
    ! [Aa: int,E_2: int,C: int,Ba: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E_2)),D) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),E_2)),C) = D ) ) ).

tff(fact_15_is__sum2sq__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(twoSqu1425020248sum2sq,X_1))
    <=> ? [A_1: int,B_3: int] : hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3)) = X_1 ) ).

tff(fact_16_Int2_Oaux1,axiom,
    ! [A_2: int,B_1: int,C_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_1) = C_1 )
     => ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_1) ) ) ).

tff(fact_17_zdiff__zmult__distrib2,axiom,
    ! [W: int,Z1: int,Z2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).

tff(fact_18_zdiff__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).

tff(fact_19_zadd__zmult__distrib2,axiom,
    ! [W: int,Z1: int,Z2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).

tff(fact_20_zadd__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).

tff(fact_21_diff__add__cancel,axiom,
    ! [A_248: rat,B_193: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_248),B_193)),B_193) = A_248 ).

tff(fact_22_diff__add__cancel,axiom,
    ! [A_248: complex,B_193: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_248),B_193)),B_193) = A_248 ).

tff(fact_23_diff__add__cancel,axiom,
    ! [A_248: real,B_193: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_248),B_193)),B_193) = A_248 ).

tff(fact_24_diff__add__cancel,axiom,
    ! [A_248: int,B_193: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_248),B_193)),B_193) = A_248 ).

tff(fact_25_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_247: rat,B_192: rat,C_101: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_247),B_192)),C_101) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_247),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_192),C_101)) ).

tff(fact_26_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_247: quickcheck_code_int,B_192: quickcheck_code_int,C_101: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_247),B_192)),C_101) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_247),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_192),C_101)) ).

tff(fact_27_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_247: complex,B_192: complex,C_101: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_247),B_192)),C_101) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_247),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_192),C_101)) ).

tff(fact_28_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_247: nat,B_192: nat,C_101: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_247),B_192)),C_101) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_247),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_192),C_101)) ).

tff(fact_29_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_247: real,B_192: real,C_101: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_247),B_192)),C_101) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_247),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_192),C_101)) ).

tff(fact_30_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_247: int,B_192: int,C_101: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_247),B_192)),C_101) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_247),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_192),C_101)) ).

tff(fact_31_add__right__imp__eq,axiom,
    ! [B_191: rat,A_246: rat,C_100: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_191),A_246) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_100),A_246) )
     => ( B_191 = C_100 ) ) ).

tff(fact_32_add__right__imp__eq,axiom,
    ! [B_191: quickcheck_code_int,A_246: quickcheck_code_int,C_100: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_191),A_246) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_100),A_246) )
     => ( B_191 = C_100 ) ) ).

tff(fact_33_add__right__imp__eq,axiom,
    ! [B_191: complex,A_246: complex,C_100: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_191),A_246) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_100),A_246) )
     => ( B_191 = C_100 ) ) ).

tff(fact_34_add__right__imp__eq,axiom,
    ! [B_191: nat,A_246: nat,C_100: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_191),A_246) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_100),A_246) )
     => ( B_191 = C_100 ) ) ).

tff(fact_35_add__right__imp__eq,axiom,
    ! [B_191: real,A_246: real,C_100: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_191),A_246) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_100),A_246) )
     => ( B_191 = C_100 ) ) ).

tff(fact_36_add__right__imp__eq,axiom,
    ! [B_191: int,A_246: int,C_100: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_191),A_246) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_100),A_246) )
     => ( B_191 = C_100 ) ) ).

tff(fact_37_add__imp__eq,axiom,
    ! [A_245: rat,B_190: rat,C_99: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_245),B_190) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_245),C_99) )
     => ( B_190 = C_99 ) ) ).

tff(fact_38_add__imp__eq,axiom,
    ! [A_245: quickcheck_code_int,B_190: quickcheck_code_int,C_99: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_245),B_190) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_245),C_99) )
     => ( B_190 = C_99 ) ) ).

tff(fact_39_add__imp__eq,axiom,
    ! [A_245: complex,B_190: complex,C_99: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_245),B_190) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_245),C_99) )
     => ( B_190 = C_99 ) ) ).

tff(fact_40_add__imp__eq,axiom,
    ! [A_245: nat,B_190: nat,C_99: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_245),B_190) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_245),C_99) )
     => ( B_190 = C_99 ) ) ).

tff(fact_41_add__imp__eq,axiom,
    ! [A_245: real,B_190: real,C_99: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_245),B_190) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_245),C_99) )
     => ( B_190 = C_99 ) ) ).

tff(fact_42_add__imp__eq,axiom,
    ! [A_245: int,B_190: int,C_99: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_245),B_190) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_245),C_99) )
     => ( B_190 = C_99 ) ) ).

tff(fact_43_add__left__imp__eq,axiom,
    ! [A_244: rat,B_189: rat,C_98: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_244),B_189) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_244),C_98) )
     => ( B_189 = C_98 ) ) ).

tff(fact_44_add__left__imp__eq,axiom,
    ! [A_244: quickcheck_code_int,B_189: quickcheck_code_int,C_98: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_244),B_189) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_244),C_98) )
     => ( B_189 = C_98 ) ) ).

tff(fact_45_add__left__imp__eq,axiom,
    ! [A_244: complex,B_189: complex,C_98: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_244),B_189) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_244),C_98) )
     => ( B_189 = C_98 ) ) ).

tff(fact_46_add__left__imp__eq,axiom,
    ! [A_244: nat,B_189: nat,C_98: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_244),B_189) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_244),C_98) )
     => ( B_189 = C_98 ) ) ).

tff(fact_47_add__left__imp__eq,axiom,
    ! [A_244: real,B_189: real,C_98: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_244),B_189) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_244),C_98) )
     => ( B_189 = C_98 ) ) ).

tff(fact_48_add__left__imp__eq,axiom,
    ! [A_244: int,B_189: int,C_98: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_244),B_189) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_244),C_98) )
     => ( B_189 = C_98 ) ) ).

tff(fact_49_add__right__cancel,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Ba),Aa) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),Aa) )
    <=> ( Ba = C ) ) ).

tff(fact_50_add__right__cancel,axiom,
    ! [Ba: quickcheck_code_int,Aa: quickcheck_code_int,C: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Ba),Aa) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),Aa) )
    <=> ( Ba = C ) ) ).

tff(fact_51_add__right__cancel,axiom,
    ! [Ba: complex,Aa: complex,C: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Ba),Aa) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),Aa) )
    <=> ( Ba = C ) ) ).

tff(fact_52_add__right__cancel,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
    <=> ( Ba = C ) ) ).

tff(fact_53_add__right__cancel,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Ba),Aa) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),Aa) )
    <=> ( Ba = C ) ) ).

tff(fact_54_add__right__cancel,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),Aa) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),Aa) )
    <=> ( Ba = C ) ) ).

tff(fact_55_add__left__cancel,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Ba) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),C) )
    <=> ( Ba = C ) ) ).

tff(fact_56_add__left__cancel,axiom,
    ! [Aa: quickcheck_code_int,Ba: quickcheck_code_int,C: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Aa),Ba) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Aa),C) )
    <=> ( Ba = C ) ) ).

tff(fact_57_add__left__cancel,axiom,
    ! [Aa: complex,Ba: complex,C: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Aa),Ba) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Aa),C) )
    <=> ( Ba = C ) ) ).

tff(fact_58_add__left__cancel,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
    <=> ( Ba = C ) ) ).

tff(fact_59_add__left__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Ba) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),C) )
    <=> ( Ba = C ) ) ).

tff(fact_60_add__left__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),C) )
    <=> ( Ba = C ) ) ).

tff(fact_61_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_243: rat,B_188: rat,C_97: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_243),B_188)),C_97) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_243),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_188),C_97)) ).

tff(fact_62_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_243: quickcheck_code_int,B_188: quickcheck_code_int,C_97: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_243),B_188)),C_97) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_243),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_188),C_97)) ).

tff(fact_63_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_243: complex,B_188: complex,C_97: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_243),B_188)),C_97) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_243),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_188),C_97)) ).

tff(fact_64_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_243: nat,B_188: nat,C_97: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_243),B_188)),C_97) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_243),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_188),C_97)) ).

tff(fact_65_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_243: real,B_188: real,C_97: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_243),B_188)),C_97) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_243),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_188),C_97)) ).

tff(fact_66_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_243: int,B_188: int,C_97: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_243),B_188)),C_97) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_243),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_188),C_97)) ).

tff(fact_67_diff__eq__diff__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

tff(fact_68_diff__eq__diff__eq,axiom,
    ! [Aa: complex,Ba: complex,C: complex,D: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Aa),Ba) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

tff(fact_69_diff__eq__diff__eq,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

tff(fact_70_diff__eq__diff__eq,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

tff(fact_71_zmult__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ).

tff(fact_72_zmult__commute,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1) ).

tff(fact_73_zadd__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ).

tff(fact_74_zadd__left__commute,axiom,
    ! [X: int,Y: int,Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z_1)) ).

tff(fact_75_zadd__commute,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1) ).

tff(fact_76_combine__common__factor,axiom,
    ! [A_242: rat,E_3: rat,B_187: rat,C_96: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_242),E_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_187),E_3)),C_96)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_242),B_187)),E_3)),C_96) ).

tff(fact_77_combine__common__factor,axiom,
    ! [A_242: quickcheck_code_int,E_3: quickcheck_code_int,B_187: quickcheck_code_int,C_96: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_242),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_187),E_3)),C_96)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_242),B_187)),E_3)),C_96) ).

tff(fact_78_combine__common__factor,axiom,
    ! [A_242: complex,E_3: complex,B_187: complex,C_96: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_242),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_187),E_3)),C_96)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_242),B_187)),E_3)),C_96) ).

tff(fact_79_combine__common__factor,axiom,
    ! [A_242: nat,E_3: nat,B_187: nat,C_96: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_242),E_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_187),E_3)),C_96)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_242),B_187)),E_3)),C_96) ).

tff(fact_80_combine__common__factor,axiom,
    ! [A_242: real,E_3: real,B_187: real,C_96: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_242),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_187),E_3)),C_96)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_242),B_187)),E_3)),C_96) ).

tff(fact_81_combine__common__factor,axiom,
    ! [A_242: int,E_3: int,B_187: int,C_96: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_242),E_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_187),E_3)),C_96)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_242),B_187)),E_3)),C_96) ).

tff(fact_82_mult__left_Oadd,axiom,
    ! [X_76: complex,Y_63: complex,Ya_4: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_76),Y_63)),Ya_4) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_76),Ya_4)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_63),Ya_4)) ).

tff(fact_83_mult__left_Oadd,axiom,
    ! [X_76: real,Y_63: real,Ya_4: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_76),Y_63)),Ya_4) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_76),Ya_4)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_63),Ya_4)) ).

tff(fact_84_mult_Oadd__left,axiom,
    ! [A_241: complex,A_240: complex,B_186: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_241),A_240)),B_186) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_241),B_186)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_240),B_186)) ).

tff(fact_85_mult_Oadd__left,axiom,
    ! [A_241: real,A_240: real,B_186: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_241),A_240)),B_186) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_241),B_186)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_240),B_186)) ).

tff(fact_86_comm__semiring__class_Odistrib,axiom,
    ! [A_239: rat,B_185: rat,C_95: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_239),B_185)),C_95) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_239),C_95)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_185),C_95)) ).

tff(fact_87_comm__semiring__class_Odistrib,axiom,
    ! [A_239: quickcheck_code_int,B_185: quickcheck_code_int,C_95: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_239),B_185)),C_95) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_239),C_95)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_185),C_95)) ).

tff(fact_88_comm__semiring__class_Odistrib,axiom,
    ! [A_239: complex,B_185: complex,C_95: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_239),B_185)),C_95) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_239),C_95)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_185),C_95)) ).

tff(fact_89_comm__semiring__class_Odistrib,axiom,
    ! [A_239: nat,B_185: nat,C_95: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_239),B_185)),C_95) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_239),C_95)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_185),C_95)) ).

tff(fact_90_comm__semiring__class_Odistrib,axiom,
    ! [A_239: real,B_185: real,C_95: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_239),B_185)),C_95) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_239),C_95)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_185),C_95)) ).

tff(fact_91_comm__semiring__class_Odistrib,axiom,
    ! [A_239: int,B_185: int,C_95: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_239),B_185)),C_95) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_239),C_95)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_185),C_95)) ).

tff(fact_92_mult__right_Oadd,axiom,
    ! [Xa_5: complex,X_75: complex,Y_62: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_5),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_75),Y_62)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_5),X_75)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_5),Y_62)) ).

tff(fact_93_mult__right_Oadd,axiom,
    ! [Xa_5: real,X_75: real,Y_62: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_5),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_75),Y_62)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_5),X_75)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_5),Y_62)) ).

tff(fact_94_mult_Oadd__right,axiom,
    ! [A_238: complex,B_184: complex,B_183: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_238),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_184),B_183)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_238),B_184)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_238),B_183)) ).

tff(fact_95_mult_Oadd__right,axiom,
    ! [A_238: real,B_184: real,B_183: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_238),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_184),B_183)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_238),B_184)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_238),B_183)) ).

tff(fact_96_mult__left_Odiff,axiom,
    ! [X_74: complex,Y_61: complex,Ya_3: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_74),Y_61)),Ya_3) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_74),Ya_3)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_61),Ya_3)) ).

tff(fact_97_mult__left_Odiff,axiom,
    ! [X_74: real,Y_61: real,Ya_3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_74),Y_61)),Ya_3) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_74),Ya_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_61),Ya_3)) ).

tff(fact_98_mult_Odiff__left,axiom,
    ! [A_237: complex,A_236: complex,B_182: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_237),A_236)),B_182) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_237),B_182)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_236),B_182)) ).

tff(fact_99_mult_Odiff__left,axiom,
    ! [A_237: real,A_236: real,B_182: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_237),A_236)),B_182) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_237),B_182)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_236),B_182)) ).

tff(fact_100_mult__right_Odiff,axiom,
    ! [Xa_4: complex,X_73: complex,Y_60: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_73),Y_60)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),X_73)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),Y_60)) ).

tff(fact_101_mult__right_Odiff,axiom,
    ! [Xa_4: real,X_73: real,Y_60: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_73),Y_60)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),X_73)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),Y_60)) ).

tff(fact_102_mult_Odiff__right,axiom,
    ! [A_235: complex,B_181: complex,B_180: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_235),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_181),B_180)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_235),B_181)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_235),B_180)) ).

tff(fact_103_mult_Odiff__right,axiom,
    ! [A_235: real,B_181: real,B_180: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_235),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_181),B_180)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_235),B_181)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_235),B_180)) ).

tff(fact_104_add__diff__add,axiom,
    ! [A_234: rat,C_94: rat,B_179: rat,D_21: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_234),C_94)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_179),D_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_234),B_179)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_94),D_21)) ).

tff(fact_105_add__diff__add,axiom,
    ! [A_234: complex,C_94: complex,B_179: complex,D_21: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_234),C_94)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_179),D_21)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_234),B_179)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_94),D_21)) ).

tff(fact_106_add__diff__add,axiom,
    ! [A_234: real,C_94: real,B_179: real,D_21: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_234),C_94)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_179),D_21)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_234),B_179)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_94),D_21)) ).

tff(fact_107_add__diff__add,axiom,
    ! [A_234: int,C_94: int,B_179: int,D_21: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_234),C_94)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_179),D_21)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_234),B_179)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_94),D_21)) ).

tff(fact_108_add__diff__cancel,axiom,
    ! [A_233: rat,B_178: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_233),B_178)),B_178) = A_233 ).

tff(fact_109_add__diff__cancel,axiom,
    ! [A_233: complex,B_178: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_233),B_178)),B_178) = A_233 ).

tff(fact_110_add__diff__cancel,axiom,
    ! [A_233: real,B_178: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_233),B_178)),B_178) = A_233 ).

tff(fact_111_add__diff__cancel,axiom,
    ! [A_233: int,B_178: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_233),B_178)),B_178) = A_233 ).

tff(fact_112_crossproduct__eq,axiom,
    ! [W_1: rat,Y_2: rat,X_1: rat,Z_2: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Y_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_113_crossproduct__eq,axiom,
    ! [W_1: complex,Y_2: complex,X_1: complex,Z_2: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Y_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_114_crossproduct__eq,axiom,
    ! [W_1: nat,Y_2: nat,X_1: nat,Z_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Z_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_115_crossproduct__eq,axiom,
    ! [W_1: real,Y_2: real,X_1: real,Z_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Y_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_116_crossproduct__eq,axiom,
    ! [W_1: int,Y_2: int,X_1: int,Z_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Y_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Z_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_232: rat,M_11: rat,B_177: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_232),M_11)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_177),M_11)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_232),B_177)),M_11) ).

tff(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_232: quickcheck_code_int,M_11: quickcheck_code_int,B_177: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_232),M_11)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_177),M_11)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_232),B_177)),M_11) ).

tff(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_232: complex,M_11: complex,B_177: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_232),M_11)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_177),M_11)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_232),B_177)),M_11) ).

tff(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_232: nat,M_11: nat,B_177: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_232),M_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_177),M_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_232),B_177)),M_11) ).

tff(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_232: real,M_11: real,B_177: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_232),M_11)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_177),M_11)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_232),B_177)),M_11) ).

tff(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_232: int,M_11: int,B_177: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_232),M_11)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_177),M_11)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_232),B_177)),M_11) ).

tff(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_231: rat,B_176: rat,C_93: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_231),B_176)),C_93) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_231),C_93)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_176),C_93)) ).

tff(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_231: quickcheck_code_int,B_176: quickcheck_code_int,C_93: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_231),B_176)),C_93) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_231),C_93)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_176),C_93)) ).

tff(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_231: complex,B_176: complex,C_93: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_231),B_176)),C_93) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_231),C_93)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_176),C_93)) ).

tff(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_231: nat,B_176: nat,C_93: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_231),B_176)),C_93) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_231),C_93)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_176),C_93)) ).

tff(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_231: real,B_176: real,C_93: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_231),B_176)),C_93) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_231),C_93)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_176),C_93)) ).

tff(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_231: int,B_176: int,C_93: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_231),B_176)),C_93) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_231),C_93)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_176),C_93)) ).

tff(fact_129_crossproduct__noteq,axiom,
    ! [C: rat,D: rat,Aa: rat,Ba: rat] :
      ( ( ( Aa != Ba )
        & ( C != D ) )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),D)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),D)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),C)) ) ) ).

tff(fact_130_crossproduct__noteq,axiom,
    ! [C: complex,D: complex,Aa: complex,Ba: complex] :
      ( ( ( Aa != Ba )
        & ( C != D ) )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ba),D)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),D)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ba),C)) ) ) ).

tff(fact_131_crossproduct__noteq,axiom,
    ! [C: nat,D: nat,Aa: nat,Ba: nat] :
      ( ( ( Aa != Ba )
        & ( C != D ) )
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),D)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),D)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),C)) ) ) ).

tff(fact_132_crossproduct__noteq,axiom,
    ! [C: real,D: real,Aa: real,Ba: real] :
      ( ( ( Aa != Ba )
        & ( C != D ) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),D)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),D)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),C)) ) ) ).

tff(fact_133_crossproduct__noteq,axiom,
    ! [C: int,D: int,Aa: int,Ba: int] :
      ( ( ( Aa != Ba )
        & ( C != D ) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),C)) ) ) ).

tff(fact_134_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_72: rat,Y_59: rat,Z_22: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_72),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_59),Z_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_72),Y_59)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_72),Z_22)) ).

tff(fact_135_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_72: quickcheck_code_int,Y_59: quickcheck_code_int,Z_22: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_72),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_59),Z_22)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_72),Y_59)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_72),Z_22)) ).

tff(fact_136_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_72: complex,Y_59: complex,Z_22: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_72),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_59),Z_22)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_72),Y_59)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_72),Z_22)) ).

tff(fact_137_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_72: nat,Y_59: nat,Z_22: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_72),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_59),Z_22)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_72),Y_59)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_72),Z_22)) ).

tff(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_72: real,Y_59: real,Z_22: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_72),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_59),Z_22)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_72),Y_59)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_72),Z_22)) ).

tff(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_72: int,Y_59: int,Z_22: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_72),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_59),Z_22)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_72),Y_59)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_72),Z_22)) ).

tff(fact_140_Pair__inject,axiom,
    ! [Aa: fun_int_bool,Ba: fun_int_bool,A_228: fun_int_bool,B_173: fun_int_bool] :
      ( ( hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,Aa),Ba) = hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_228),B_173) )
     => ~ ( ( Aa = A_228 )
         => ( Ba != B_173 ) ) ) ).

tff(fact_141_Pair__inject,axiom,
    ! [A_230: product_prod_int_int,B_175: product_prod_int_int,A_229: product_prod_int_int,B_174: product_prod_int_int] :
      ( ( hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_230),B_175) = hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_142_Pair__inject,axiom,
    ! [A_230: quickcheck_code_int,B_175: quickcheck_code_int,A_229: quickcheck_code_int,B_174: quickcheck_code_int] :
      ( ( hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,A_230),B_175) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_143_Pair__inject,axiom,
    ! [A_230: code_code_numeral,B_175: code_code_numeral,A_229: code_code_numeral,B_174: code_code_numeral] :
      ( ( hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,A_230),B_175) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_144_Pair__inject,axiom,
    ! [A_230: real,B_175: real,A_229: real,B_174: real] :
      ( ( hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_230),B_175) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_145_Pair__inject,axiom,
    ! [A_230: product_prod_nat_nat,B_175: product_prod_nat_nat,A_229: product_prod_nat_nat,B_174: product_prod_nat_nat] :
      ( ( hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_230),B_175) = hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_146_Pair__inject,axiom,
    ! [A_230: nat,B_175: nat,A_229: nat,B_174: nat] :
      ( ( hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A_230),B_175) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_147_Pair__inject,axiom,
    ! [A_230: int,B_175: product_prod_int_int,A_229: int,B_174: product_prod_int_int] :
      ( ( hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_230),B_175) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_148_Pair__inject,axiom,
    ! [A_230: int,B_175: int,A_229: int,B_174: int] :
      ( ( hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_230),B_175) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_229),B_174) )
     => ~ ( ( A_230 = A_229 )
         => ( B_175 != B_174 ) ) ) ).

tff(fact_149_Pair__eq,axiom,
    ! [Aa: fun_int_bool,Ba: fun_int_bool,A_228: fun_int_bool,B_173: fun_int_bool] :
      ( ( hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,Aa),Ba) = hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_150_Pair__eq,axiom,
    ! [Aa: product_prod_int_int,Ba: product_prod_int_int,A_228: product_prod_int_int,B_173: product_prod_int_int] :
      ( ( hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,Aa),Ba) = hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_151_Pair__eq,axiom,
    ! [Aa: quickcheck_code_int,Ba: quickcheck_code_int,A_228: quickcheck_code_int,B_173: quickcheck_code_int] :
      ( ( hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,Aa),Ba) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_152_Pair__eq,axiom,
    ! [Aa: code_code_numeral,Ba: code_code_numeral,A_228: code_code_numeral,B_173: code_code_numeral] :
      ( ( hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,Aa),Ba) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_153_Pair__eq,axiom,
    ! [Aa: real,Ba: real,A_228: real,B_173: real] :
      ( ( hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_154_Pair__eq,axiom,
    ! [Aa: product_prod_nat_nat,Ba: product_prod_nat_nat,A_228: product_prod_nat_nat,B_173: product_prod_nat_nat] :
      ( ( hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,Aa),Ba) = hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_155_Pair__eq,axiom,
    ! [Aa: nat,Ba: nat,A_228: nat,B_173: nat] :
      ( ( hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),Ba) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_156_Pair__eq,axiom,
    ! [Aa: int,Ba: product_prod_int_int,A_228: int,B_173: product_prod_int_int] :
      ( ( hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Aa),Ba) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_157_Pair__eq,axiom,
    ! [Aa: int,Ba: int,A_228: int,B_173: int] :
      ( ( hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_228),B_173) )
    <=> ( ( Aa = A_228 )
        & ( Ba = B_173 ) ) ) ).

tff(fact_158_split__paired__All,axiom,
    ! [P_1: fun_Pr4007113l_bool] :
      ( ! [X1: produc1450482951t_bool] : hBOOL(hAPP_P880235623l_bool(P_1,X1))
    <=> ! [A_1: fun_int_bool,B_3: fun_int_bool] : hBOOL(hAPP_P880235623l_bool(P_1,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_3))) ) ).

tff(fact_159_split__paired__All,axiom,
    ! [P_1: fun_Pr1333588703t_bool] :
      ( ! [X1: produc1137372701nt_int] : hBOOL(hAPP_P62333565t_bool(P_1,X1))
    <=> ! [A_1: product_prod_int_int,B_3: product_prod_int_int] : hBOOL(hAPP_P62333565t_bool(P_1,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_1),B_3))) ) ).

tff(fact_160_split__paired__All,axiom,
    ! [P_1: fun_Pr610342249t_bool] :
      ( ! [X1: produc167071911de_int] : hBOOL(hAPP_P1709355527t_bool(P_1,X1))
    <=> ! [A_1: quickcheck_code_int,B_3: quickcheck_code_int] : hBOOL(hAPP_P1709355527t_bool(P_1,hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,A_1),B_3))) ) ).

tff(fact_161_split__paired__All,axiom,
    ! [P_1: fun_Pr854501993l_bool] :
      ( ! [X1: produc1359518119umeral] : hBOOL(hAPP_P1164677895l_bool(P_1,X1))
    <=> ! [A_1: code_code_numeral,B_3: code_code_numeral] : hBOOL(hAPP_P1164677895l_bool(P_1,hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,A_1),B_3))) ) ).

tff(fact_162_split__paired__All,axiom,
    ! [P_1: fun_Pr1127657839l_bool] :
      ( ! [X1: produc914805421l_real] : hBOOL(hAPP_P1333854989l_bool(P_1,X1))
    <=> ! [A_1: real,B_3: real] : hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3))) ) ).

tff(fact_163_split__paired__All,axiom,
    ! [P_1: fun_Pr587612383t_bool] :
      ( ! [X1: produc1322466333at_nat] : hBOOL(hAPP_P1463840893t_bool(P_1,X1))
    <=> ! [A_1: product_prod_nat_nat,B_3: product_prod_nat_nat] : hBOOL(hAPP_P1463840893t_bool(P_1,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_1),B_3))) ) ).

tff(fact_164_split__paired__All,axiom,
    ! [P_1: fun_Pr1927655017t_bool] :
      ( ! [X1: product_prod_nat_nat] : hBOOL(hAPP_P1555980039t_bool(P_1,X1))
    <=> ! [A_1: nat,B_3: nat] : hBOOL(hAPP_P1555980039t_bool(P_1,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A_1),B_3))) ) ).

tff(fact_165_split__paired__All,axiom,
    ! [P_1: fun_Pr1503267378t_bool] :
      ( ! [X1: produc393999548nt_int] : hBOOL(hAPP_P178748804t_bool(P_1,X1))
    <=> ! [A_1: int,B_3: product_prod_int_int] : hBOOL(hAPP_P178748804t_bool(P_1,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_1),B_3))) ) ).

tff(fact_166_split__paired__All,axiom,
    ! [P_1: fun_Pr974702441t_bool] :
      ( ! [X1: product_prod_int_int] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
    <=> ! [A_1: int,B_3: int] : hBOOL(hAPP_P603027463t_bool(P_1,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3))) ) ).

tff(fact_167_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_227: rat,B_172: rat,C_92: rat,D_20: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_227),B_172)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_92),D_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_227),C_92)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_172),D_20)) ).

tff(fact_168_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_227: quickcheck_code_int,B_172: quickcheck_code_int,C_92: quickcheck_code_int,D_20: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_227),B_172)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_92),D_20)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_227),C_92)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_172),D_20)) ).

tff(fact_169_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_227: complex,B_172: complex,C_92: complex,D_20: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_227),B_172)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_92),D_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_227),C_92)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_172),D_20)) ).

tff(fact_170_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_227: nat,B_172: nat,C_92: nat,D_20: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_227),B_172)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_92),D_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_227),C_92)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_172),D_20)) ).

tff(fact_171_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_227: real,B_172: real,C_92: real,D_20: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_227),B_172)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_92),D_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_227),C_92)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_172),D_20)) ).

tff(fact_172_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_227: int,B_172: int,C_92: int,D_20: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_227),B_172)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_92),D_20)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_227),C_92)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_172),D_20)) ).

tff(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: rat,B_171: rat,C_91: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),B_171)),C_91) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),C_91)),B_171) ).

tff(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: quickcheck_code_int,B_171: quickcheck_code_int,C_91: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),B_171)),C_91) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),C_91)),B_171) ).

tff(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: complex,B_171: complex,C_91: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),B_171)),C_91) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),C_91)),B_171) ).

tff(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: nat,B_171: nat,C_91: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),B_171)),C_91) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),C_91)),B_171) ).

tff(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: real,B_171: real,C_91: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),B_171)),C_91) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),C_91)),B_171) ).

tff(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: int,B_171: int,C_91: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),B_171)),C_91) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),C_91)),B_171) ).

tff(fact_179_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_225: rat,B_170: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_225),B_170) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_170),A_225) ).

tff(fact_180_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_225: quickcheck_code_int,B_170: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_225),B_170) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_170),A_225) ).

tff(fact_181_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_225: complex,B_170: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_225),B_170) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_170),A_225) ).

tff(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_225: nat,B_170: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_225),B_170) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_170),A_225) ).

tff(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_225: real,B_170: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_225),B_170) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_170),A_225) ).

tff(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_225: int,B_170: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_225),B_170) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_170),A_225) ).

tff(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx_6: rat,Rx_6: rat,Ry_4: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),Ry_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ry_4)) ).

tff(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx_6: quickcheck_code_int,Rx_6: quickcheck_code_int,Ry_4: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ry_4)) ).

tff(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx_6: complex,Rx_6: complex,Ry_4: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ry_4)) ).

tff(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx_6: nat,Rx_6: nat,Ry_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ry_4)) ).

tff(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx_6: real,Rx_6: real,Ry_4: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ry_4)) ).

tff(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx_6: int,Rx_6: int,Ry_4: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ry_4)) ).

tff(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_5: rat,Rx_5: rat,Ry_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),Ry_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Rx_5)),Ry_3) ).

tff(fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_5: quickcheck_code_int,Rx_5: quickcheck_code_int,Ry_3: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Rx_5)),Ry_3) ).

tff(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_5: complex,Rx_5: complex,Ry_3: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Rx_5)),Ry_3) ).

tff(fact_194_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_5: nat,Rx_5: nat,Ry_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Rx_5)),Ry_3) ).

tff(fact_195_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_5: real,Rx_5: real,Ry_3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Rx_5)),Ry_3) ).

tff(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_5: int,Rx_5: int,Ry_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Rx_5)),Ry_3) ).

tff(fact_197_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_4: rat,Ly_4: rat,Rx_4: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),Ly_4)),Rx_4) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_4),Rx_4)) ).

tff(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_4: quickcheck_code_int,Ly_4: quickcheck_code_int,Rx_4: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_4)),Rx_4) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Rx_4)) ).

tff(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_4: complex,Ly_4: complex,Rx_4: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_4)),Rx_4) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Rx_4)) ).

tff(fact_200_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_4: nat,Ly_4: nat,Rx_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_4)),Rx_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Rx_4)) ).

tff(fact_201_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_4: real,Ly_4: real,Rx_4: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_4)),Rx_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Rx_4)) ).

tff(fact_202_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_4: int,Ly_4: int,Rx_4: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_4)),Rx_4) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Rx_4)) ).

tff(fact_203_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: rat,Ly_3: rat,Rx_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Ly_3)),Rx_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Rx_3)),Ly_3) ).

tff(fact_204_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: quickcheck_code_int,Ly_3: quickcheck_code_int,Rx_3: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_3)),Rx_3) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Rx_3)),Ly_3) ).

tff(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: complex,Ly_3: complex,Rx_3: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_3)),Rx_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_3) ).

tff(fact_206_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: nat,Ly_3: nat,Rx_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_3)),Rx_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_3) ).

tff(fact_207_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: real,Ly_3: real,Rx_3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_3)),Rx_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Rx_3)),Ly_3) ).

tff(fact_208_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: int,Ly_3: int,Rx_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_3)),Rx_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_3) ).

tff(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_2: rat,Ly_2: rat,Rx_2: rat,Ry_2: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_2),Ry_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_2),Ry_2))) ).

tff(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_2: quickcheck_code_int,Ly_2: quickcheck_code_int,Rx_2: quickcheck_code_int,Ry_2: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_2),Ry_2)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_2),Ry_2))) ).

tff(fact_211_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_2: complex,Ly_2: complex,Rx_2: complex,Ry_2: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_2),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_2),Ry_2))) ).

tff(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_2))) ).

tff(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_2: real,Ly_2: real,Rx_2: real,Ry_2: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_2),Ry_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_2),Ry_2))) ).

tff(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_2))) ).

tff(fact_215_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_1: rat,Ly_1: rat,Rx_1: rat,Ry_1: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Ly_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),Ry_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Ly_1)),Ry_1)) ).

tff(fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_1: quickcheck_code_int,Ly_1: quickcheck_code_int,Rx_1: quickcheck_code_int,Ry_1: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Ly_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Ly_1)),Ry_1)) ).

tff(fact_217_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_1: complex,Ly_1: complex,Rx_1: complex,Ry_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Ly_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Ly_1)),Ry_1)) ).

tff(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Ly_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Ly_1)),Ry_1)) ).

tff(fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_1: real,Ly_1: real,Rx_1: real,Ry_1: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Ly_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Ly_1)),Ry_1)) ).

tff(fact_220_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Ly_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Ly_1)),Ry_1)) ).

tff(fact_221_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx: rat,Ly: rat,Rx: rat,Ry: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ly)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Rx)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Ry)) ).

tff(fact_222_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx: quickcheck_code_int,Ly: quickcheck_code_int,Rx: quickcheck_code_int,Ry: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ly)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Rx)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Ry)) ).

tff(fact_223_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx: complex,Ly: complex,Rx: complex,Ry: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ly)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Rx)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Ry)) ).

tff(fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ly)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Rx)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Ry)) ).

tff(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx: real,Ly: real,Rx: real,Ry: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ly)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Rx)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Ry)) ).

tff(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx: int,Ly: int,Rx: int,Ry: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ly)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Rx)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Ry)) ).

tff(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_224: rat,C_90: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_224),C_90) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_90),A_224) ).

tff(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_224: quickcheck_code_int,C_90: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_224),C_90) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_90),A_224) ).

tff(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_224: complex,C_90: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_224),C_90) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_90),A_224) ).

tff(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_224: nat,C_90: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_224),C_90) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_90),A_224) ).

tff(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_224: real,C_90: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_224),C_90) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_90),A_224) ).

tff(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_224: int,C_90: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_224),C_90) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_90),A_224) ).

tff(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_223: rat,C_89: rat,D_19: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_223),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_89),D_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_89),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_223),D_19)) ).

tff(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_223: quickcheck_code_int,C_89: quickcheck_code_int,D_19: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_223),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_89),D_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_89),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_223),D_19)) ).

tff(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_223: complex,C_89: complex,D_19: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_223),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_89),D_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_89),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_223),D_19)) ).

tff(fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_223: nat,C_89: nat,D_19: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_89),D_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_89),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_223),D_19)) ).

tff(fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_223: real,C_89: real,D_19: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_223),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_89),D_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_89),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_223),D_19)) ).

tff(fact_238_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_223: int,C_89: int,D_19: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_223),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_89),D_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_89),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_223),D_19)) ).

tff(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_222: rat,C_88: rat,D_18: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_222),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_88),D_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_222),C_88)),D_18) ).

tff(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_222: quickcheck_code_int,C_88: quickcheck_code_int,D_18: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_222),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_88),D_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_222),C_88)),D_18) ).

tff(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_222: complex,C_88: complex,D_18: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_222),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_88),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_222),C_88)),D_18) ).

tff(fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_222: nat,C_88: nat,D_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_222),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_88),D_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_222),C_88)),D_18) ).

tff(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_222: real,C_88: real,D_18: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_222),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_88),D_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_222),C_88)),D_18) ).

tff(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_222: int,C_88: int,D_18: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_222),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_88),D_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_222),C_88)),D_18) ).

tff(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_221: rat,B_169: rat,C_87: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_221),B_169)),C_87) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_221),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_169),C_87)) ).

tff(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_221: quickcheck_code_int,B_169: quickcheck_code_int,C_87: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_221),B_169)),C_87) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_221),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_169),C_87)) ).

tff(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_221: complex,B_169: complex,C_87: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_221),B_169)),C_87) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_221),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_169),C_87)) ).

tff(fact_248_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_221: nat,B_169: nat,C_87: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_221),B_169)),C_87) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_221),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_169),C_87)) ).

tff(fact_249_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_221: real,B_169: real,C_87: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_221),B_169)),C_87) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_221),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_169),C_87)) ).

tff(fact_250_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_221: int,B_169: int,C_87: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_221),B_169)),C_87) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_221),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_169),C_87)) ).

tff(fact_251_split__paired__Ex,axiom,
    ! [P_1: fun_Pr4007113l_bool] :
      ( ? [X1: produc1450482951t_bool] : hBOOL(hAPP_P880235623l_bool(P_1,X1))
    <=> ? [A_1: fun_int_bool,B_3: fun_int_bool] : hBOOL(hAPP_P880235623l_bool(P_1,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_3))) ) ).

tff(fact_252_split__paired__Ex,axiom,
    ! [P_1: fun_Pr1333588703t_bool] :
      ( ? [X1: produc1137372701nt_int] : hBOOL(hAPP_P62333565t_bool(P_1,X1))
    <=> ? [A_1: product_prod_int_int,B_3: product_prod_int_int] : hBOOL(hAPP_P62333565t_bool(P_1,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_1),B_3))) ) ).

tff(fact_253_split__paired__Ex,axiom,
    ! [P_1: fun_Pr610342249t_bool] :
      ( ? [X1: produc167071911de_int] : hBOOL(hAPP_P1709355527t_bool(P_1,X1))
    <=> ? [A_1: quickcheck_code_int,B_3: quickcheck_code_int] : hBOOL(hAPP_P1709355527t_bool(P_1,hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,A_1),B_3))) ) ).

tff(fact_254_split__paired__Ex,axiom,
    ! [P_1: fun_Pr854501993l_bool] :
      ( ? [X1: produc1359518119umeral] : hBOOL(hAPP_P1164677895l_bool(P_1,X1))
    <=> ? [A_1: code_code_numeral,B_3: code_code_numeral] : hBOOL(hAPP_P1164677895l_bool(P_1,hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,A_1),B_3))) ) ).

tff(fact_255_split__paired__Ex,axiom,
    ! [P_1: fun_Pr1127657839l_bool] :
      ( ? [X1: produc914805421l_real] : hBOOL(hAPP_P1333854989l_bool(P_1,X1))
    <=> ? [A_1: real,B_3: real] : hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3))) ) ).

tff(fact_256_split__paired__Ex,axiom,
    ! [P_1: fun_Pr587612383t_bool] :
      ( ? [X1: produc1322466333at_nat] : hBOOL(hAPP_P1463840893t_bool(P_1,X1))
    <=> ? [A_1: product_prod_nat_nat,B_3: product_prod_nat_nat] : hBOOL(hAPP_P1463840893t_bool(P_1,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_1),B_3))) ) ).

tff(fact_257_split__paired__Ex,axiom,
    ! [P_1: fun_Pr1927655017t_bool] :
      ( ? [X1: product_prod_nat_nat] : hBOOL(hAPP_P1555980039t_bool(P_1,X1))
    <=> ? [A_1: nat,B_3: nat] : hBOOL(hAPP_P1555980039t_bool(P_1,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A_1),B_3))) ) ).

tff(fact_258_split__paired__Ex,axiom,
    ! [P_1: fun_Pr1503267378t_bool] :
      ( ? [X1: produc393999548nt_int] : hBOOL(hAPP_P178748804t_bool(P_1,X1))
    <=> ? [A_1: int,B_3: product_prod_int_int] : hBOOL(hAPP_P178748804t_bool(P_1,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_1),B_3))) ) ).

tff(fact_259_split__paired__Ex,axiom,
    ! [P_1: fun_Pr974702441t_bool] :
      ( ? [X1: product_prod_int_int] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
    <=> ? [A_1: int,B_3: int] : hBOOL(hAPP_P603027463t_bool(P_1,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3))) ) ).

tff(fact_260_prod__induct3,axiom,
    ! [X_1: produc1137372701nt_int,P_1: fun_Pr1333588703t_bool] :
      ( ! [A_1: product_prod_int_int,B_3: int,C_3: int] : hBOOL(hAPP_P62333565t_bool(P_1,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,B_3),C_3))))
     => hBOOL(hAPP_P62333565t_bool(P_1,X_1)) ) ).

tff(fact_261_prod__induct3,axiom,
    ! [X_1: produc1322466333at_nat,P_1: fun_Pr587612383t_bool] :
      ( ! [A_1: product_prod_nat_nat,B_3: nat,C_3: nat] : hBOOL(hAPP_P1463840893t_bool(P_1,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_3),C_3))))
     => hBOOL(hAPP_P1463840893t_bool(P_1,X_1)) ) ).

tff(fact_262_prod__induct3,axiom,
    ! [X_1: produc393999548nt_int,P_1: fun_Pr1503267378t_bool] :
      ( ! [A_1: int,B_3: int,C_3: int] : hBOOL(hAPP_P178748804t_bool(P_1,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,B_3),C_3))))
     => hBOOL(hAPP_P178748804t_bool(P_1,X_1)) ) ).

tff(fact_263_prod__cases3,axiom,
    ! [Y_58: produc1137372701nt_int] :
      ~ ! [A_1: product_prod_int_int,B_3: int,C_3: int] : Y_58 != hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,B_3),C_3)) ).

tff(fact_264_prod__cases3,axiom,
    ! [Y_58: produc1322466333at_nat] :
      ~ ! [A_1: product_prod_nat_nat,B_3: nat,C_3: nat] : Y_58 != hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_3),C_3)) ).

tff(fact_265_prod__cases3,axiom,
    ! [Y_58: produc393999548nt_int] :
      ~ ! [A_1: int,B_3: int,C_3: int] : Y_58 != hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,B_3),C_3)) ).

tff(fact_266_PairE,axiom,
    ! [Pa: produc1450482951t_bool] :
      ~ ! [X_2: fun_int_bool,Y_1: fun_int_bool] : Pa != hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,X_2),Y_1) ).

tff(fact_267_PairE,axiom,
    ! [P_7: produc1137372701nt_int] :
      ~ ! [X_2: product_prod_int_int,Y_1: product_prod_int_int] : P_7 != hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_2),Y_1) ).

tff(fact_268_PairE,axiom,
    ! [P_7: produc167071911de_int] :
      ~ ! [X_2: quickcheck_code_int,Y_1: quickcheck_code_int] : P_7 != hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,X_2),Y_1) ).

tff(fact_269_PairE,axiom,
    ! [P_7: produc1359518119umeral] :
      ~ ! [X_2: code_code_numeral,Y_1: code_code_numeral] : P_7 != hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,X_2),Y_1) ).

tff(fact_270_PairE,axiom,
    ! [P_7: produc914805421l_real] :
      ~ ! [X_2: real,Y_1: real] : P_7 != hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,X_2),Y_1) ).

tff(fact_271_PairE,axiom,
    ! [P_7: produc1322466333at_nat] :
      ~ ! [X_2: product_prod_nat_nat,Y_1: product_prod_nat_nat] : P_7 != hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,X_2),Y_1) ).

tff(fact_272_PairE,axiom,
    ! [P_7: product_prod_nat_nat] :
      ~ ! [X_2: nat,Y_1: nat] : P_7 != hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_2),Y_1) ).

tff(fact_273_PairE,axiom,
    ! [P_7: produc393999548nt_int] :
      ~ ! [X_2: int,Y_1: product_prod_int_int] : P_7 != hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,X_2),Y_1) ).

tff(fact_274_PairE,axiom,
    ! [P_7: product_prod_int_int] :
      ~ ! [X_2: int,Y_1: int] : P_7 != hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,X_2),Y_1) ).

tff(fact_275_mem__def,axiom,
    ! [X_1: produc1450482951t_bool,A: fun_Pr4007113l_bool] :
      ( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,X_1),A))
    <=> hBOOL(hAPP_P880235623l_bool(A,X_1)) ) ).

tff(fact_276_mem__def,axiom,
    ! [X_1: produc1137372701nt_int,A: fun_Pr1333588703t_bool] :
      ( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,X_1),A))
    <=> hBOOL(hAPP_P62333565t_bool(A,X_1)) ) ).

tff(fact_277_mem__def,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).

tff(fact_278_mem__def,axiom,
    ! [X_1: fun_int_bool,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
    <=> hBOOL(hAPP_f448129468l_bool(A,X_1)) ) ).

tff(fact_279_mem__def,axiom,
    ! [X_1: produc1322466333at_nat,A: fun_Pr587612383t_bool] :
      ( hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,X_1),A))
    <=> hBOOL(hAPP_P1463840893t_bool(A,X_1)) ) ).

tff(fact_280_mem__def,axiom,
    ! [X_1: real,A: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),A))
    <=> hBOOL(hAPP_real_bool(A,X_1)) ) ).

tff(fact_281_mem__def,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
    <=> hBOOL(hAPP_int_bool(A,X_1)) ) ).

tff(fact_282_Collect__def,axiom,
    ! [P_1: fun_Pr1927655017t_bool] : collec1979865426at_nat(P_1) = P_1 ).

tff(fact_283_Collect__def,axiom,
    ! [P_1: fun_real_bool] : collect_real(P_1) = P_1 ).

tff(fact_284_Collect__def,axiom,
    ! [P_1: fun_Pr974702441t_bool] : collec1347809874nt_int(P_1) = P_1 ).

tff(fact_285_Collect__def,axiom,
    ! [P_1: fun_Pr1333588703t_bool] : collec50511176nt_int(P_1) = P_1 ).

tff(fact_286_Collect__def,axiom,
    ! [P_1: fun_nat_bool] : collect_nat(P_1) = P_1 ).

tff(fact_287_Collect__def,axiom,
    ! [P_1: fun_int_bool] : collect_int(P_1) = P_1 ).

tff(fact_288_prod_Oexhaust,axiom,
    ! [Y_2: produc1450482951t_bool] :
      ~ ! [A_1: fun_int_bool,B_3: fun_int_bool] : Y_2 != hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_3) ).

tff(fact_289_prod_Oexhaust,axiom,
    ! [Y_57: produc1137372701nt_int] :
      ~ ! [A_1: product_prod_int_int,B_3: product_prod_int_int] : Y_57 != hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,A_1),B_3) ).

tff(fact_290_prod_Oexhaust,axiom,
    ! [Y_57: produc167071911de_int] :
      ~ ! [A_1: quickcheck_code_int,B_3: quickcheck_code_int] : Y_57 != hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,A_1),B_3) ).

tff(fact_291_prod_Oexhaust,axiom,
    ! [Y_57: produc1359518119umeral] :
      ~ ! [A_1: code_code_numeral,B_3: code_code_numeral] : Y_57 != hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,A_1),B_3) ).

tff(fact_292_prod_Oexhaust,axiom,
    ! [Y_57: produc914805421l_real] :
      ~ ! [A_1: real,B_3: real] : Y_57 != hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3) ).

tff(fact_293_prod_Oexhaust,axiom,
    ! [Y_57: produc1322466333at_nat] :
      ~ ! [A_1: product_prod_nat_nat,B_3: product_prod_nat_nat] : Y_57 != hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,A_1),B_3) ).

tff(fact_294_prod_Oexhaust,axiom,
    ! [Y_57: product_prod_nat_nat] :
      ~ ! [A_1: nat,B_3: nat] : Y_57 != hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A_1),B_3) ).

tff(fact_295_prod_Oexhaust,axiom,
    ! [Y_57: produc393999548nt_int] :
      ~ ! [A_1: int,B_3: product_prod_int_int] : Y_57 != hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,A_1),B_3) ).

tff(fact_296_prod_Oexhaust,axiom,
    ! [Y_57: product_prod_int_int] :
      ~ ! [A_1: int,B_3: int] : Y_57 != hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3) ).

tff(fact_297_curryI,axiom,
    ! [F: fun_Pr4007113l_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_P880235623l_bool(F,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,Aa),Ba)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(produc1775888740l_bool(F),Aa),Ba)) ) ).

tff(fact_298_curryI,axiom,
    ! [F: fun_Pr1333588703t_bool,Aa: product_prod_int_int,Ba: product_prod_int_int] :
      ( hBOOL(hAPP_P62333565t_bool(F,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,Aa),Ba)))
     => hBOOL(hAPP_P603027463t_bool(hAPP_P989584703t_bool(produc353512698t_bool(F),Aa),Ba)) ) ).

tff(fact_299_curryI,axiom,
    ! [F: fun_Pr610342249t_bool,Aa: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_P1709355527t_bool(F,hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,Aa),Ba)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(produc2013771390t_bool(F),Aa),Ba)) ) ).

tff(fact_300_curryI,axiom,
    ! [F: fun_Pr854501993l_bool,Aa: code_code_numeral,Ba: code_code_numeral] :
      ( hBOOL(hAPP_P1164677895l_bool(F,hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,Aa),Ba)))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(produc609372286l_bool(F),Aa),Ba)) ) ).

tff(fact_301_curryI,axiom,
    ! [F: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_P1333854989l_bool(F,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(produc748992778l_bool(F),Aa),Ba)) ) ).

tff(fact_302_curryI,axiom,
    ! [F: fun_Pr587612383t_bool,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1463840893t_bool(F,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,Aa),Ba)))
     => hBOOL(hAPP_P1555980039t_bool(hAPP_P705731903t_bool(produc1319041786t_bool(F),Aa),Ba)) ) ).

tff(fact_303_curryI,axiom,
    ! [F: fun_Pr1927655017t_bool,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(F,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),Ba)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(produc2091540094t_bool(F),Aa),Ba)) ) ).

tff(fact_304_curryI,axiom,
    ! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
      ( hBOOL(hAPP_P178748804t_bool(F,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Aa),Ba)))
     => hBOOL(hAPP_P603027463t_bool(hAPP_i119638232t_bool(produc97574839t_bool(F),Aa),Ba)) ) ).

tff(fact_305_curryI,axiom,
    ! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
      ( hBOOL(hAPP_P603027463t_bool(F,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(produc262399358t_bool(F),Aa),Ba)) ) ).

tff(fact_306_le__add__iff1,axiom,
    ! [Aa: rat,E_2: rat,C: rat,Ba: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba)),E_2)),C)),D)) ) ).

tff(fact_307_le__add__iff1,axiom,
    ! [Aa: real,E_2: real,C: real,Ba: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba)),E_2)),C)),D)) ) ).

tff(fact_308_le__add__iff1,axiom,
    ! [Aa: int,E_2: int,C: int,Ba: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),E_2)),C)),D)) ) ).

tff(fact_309_le__add__iff2,axiom,
    ! [Aa: rat,E_2: rat,C: rat,Ba: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Ba),Aa)),E_2)),D))) ) ).

tff(fact_310_le__add__iff2,axiom,
    ! [Aa: real,E_2: real,C: real,Ba: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)),E_2)),D))) ) ).

tff(fact_311_le__add__iff2,axiom,
    ! [Aa: int,E_2: int,C: int,Ba: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),Aa)),E_2)),D))) ) ).

tff(fact_312_real__squared__diff__one__factored,axiom,
    ! [X_71: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_71),X_71)),one_one_rat) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_71),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_71),one_one_rat)) ).

tff(fact_313_real__squared__diff__one__factored,axiom,
    ! [X_71: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_71),X_71)),one_one_complex) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_71),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_71),one_one_complex)) ).

tff(fact_314_real__squared__diff__one__factored,axiom,
    ! [X_71: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_71),X_71)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_71),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_71),one_one_real)) ).

tff(fact_315_real__squared__diff__one__factored,axiom,
    ! [X_71: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_71),X_71)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_71),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_71),one_one_int)) ).

tff(fact_316_less__add__iff1,axiom,
    ! [Aa: rat,E_2: rat,C: rat,Ba: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba)),E_2)),C)),D)) ) ).

tff(fact_317_less__add__iff1,axiom,
    ! [Aa: real,E_2: real,C: real,Ba: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba)),E_2)),C)),D)) ) ).

tff(fact_318_less__add__iff1,axiom,
    ! [Aa: int,E_2: int,C: int,Ba: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),E_2)),C)),D)) ) ).

tff(fact_319_less__add__iff2,axiom,
    ! [Aa: rat,E_2: rat,C: rat,Ba: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Ba),Aa)),E_2)),D))) ) ).

tff(fact_320_less__add__iff2,axiom,
    ! [Aa: real,E_2: real,C: real,Ba: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)),E_2)),D))) ) ).

tff(fact_321_less__add__iff2,axiom,
    ! [Aa: int,E_2: int,C: int,Ba: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),Aa)),E_2)),D))) ) ).

tff(fact_322_add__number__of__diff1,axiom,
    ! [V_25: int,W_17: int,C_86: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_25)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_17)),C_86)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),W_17))),C_86) ).

tff(fact_323_add__number__of__diff1,axiom,
    ! [V_25: int,W_17: int,C_86: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_25)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_17)),C_86)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),W_17))),C_86) ).

tff(fact_324_add__number__of__diff1,axiom,
    ! [V_25: int,W_17: int,C_86: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_25)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_17)),C_86)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),W_17))),C_86) ).

tff(fact_325_add__number__of__diff1,axiom,
    ! [V_25: int,W_17: int,C_86: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_25)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_17)),C_86)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),W_17))),C_86) ).

tff(fact_326_in__rel__def,axiom,
    ! [R_4: fun_Pr4007113l_bool,X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(in_rel512175208t_bool(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,X_1),Y_2)),R_4)) ) ).

tff(fact_327_in__rel__def,axiom,
    ! [R_4: fun_Pr1333588703t_bool,X_1: product_prod_int_int,Y_2: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(hAPP_P989584703t_bool(in_rel682537726nt_int(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_1),Y_2)),R_4)) ) ).

tff(fact_328_in__rel__def,axiom,
    ! [R_4: fun_Pr587612383t_bool,X_1: product_prod_nat_nat,Y_2: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(hAPP_P705731903t_bool(in_rel293241086at_nat(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,X_1),Y_2)),R_4)) ) ).

tff(fact_329_in__rel__def,axiom,
    ! [R_4: fun_Pr610342249t_bool,X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(in_rel1707483778de_int(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f222989233l_bool(hAPP_P1879777545l_bool(member2092545300de_int,hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,X_1),Y_2)),R_4)) ) ).

tff(fact_330_in__rel__def,axiom,
    ! [R_4: fun_Pr854501993l_bool,X_1: code_code_numeral,Y_2: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(in_rel1251239042umeral(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f251496625l_bool(hAPP_P2045320969l_bool(member1246959636umeral,hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,X_1),Y_2)),R_4)) ) ).

tff(fact_331_in__rel__def,axiom,
    ! [R_4: fun_Pr1127657839l_bool,X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(in_rel_real_real(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f46817975l_bool(hAPP_P1946311189l_bool(member1344826522l_real,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,X_1),Y_2)),R_4)) ) ).

tff(fact_332_in__rel__def,axiom,
    ! [R_4: fun_Pr1927655017t_bool,X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(in_rel_nat_nat(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f879361201l_bool(hAPP_P1251024649l_bool(member232255508at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_1),Y_2)),R_4)) ) ).

tff(fact_333_in__rel__def,axiom,
    ! [R_4: fun_Pr1503267378t_bool,X_1: int,Y_2: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(hAPP_i119638232t_bool(in_rel1239871883nt_int(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f928428050l_bool(hAPP_P2074576781l_bool(member791087631nt_int,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,X_1),Y_2)),R_4)) ) ).

tff(fact_334_in__rel__def,axiom,
    ! [R_4: fun_Pr974702441t_bool,X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(in_rel_int_int(R_4),X_1),Y_2))
    <=> hBOOL(hAPP_f851219889l_bool(hAPP_P1268961545l_bool(member1747683604nt_int,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,X_1),Y_2)),R_4)) ) ).

tff(fact_335_curryE,axiom,
    ! [F: fun_Pr4007113l_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(produc1775888740l_bool(F),Aa),Ba))
     => hBOOL(hAPP_P880235623l_bool(F,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,Aa),Ba))) ) ).

tff(fact_336_curryE,axiom,
    ! [F: fun_Pr1333588703t_bool,Aa: product_prod_int_int,Ba: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(hAPP_P989584703t_bool(produc353512698t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P62333565t_bool(F,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,Aa),Ba))) ) ).

tff(fact_337_curryE,axiom,
    ! [F: fun_Pr610342249t_bool,Aa: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(produc2013771390t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1709355527t_bool(F,hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,Aa),Ba))) ) ).

tff(fact_338_curryE,axiom,
    ! [F: fun_Pr854501993l_bool,Aa: code_code_numeral,Ba: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(produc609372286l_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1164677895l_bool(F,hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,Aa),Ba))) ) ).

tff(fact_339_curryE,axiom,
    ! [F: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(produc748992778l_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1333854989l_bool(F,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba))) ) ).

tff(fact_340_curryE,axiom,
    ! [F: fun_Pr587612383t_bool,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(hAPP_P705731903t_bool(produc1319041786t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1463840893t_bool(F,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,Aa),Ba))) ) ).

tff(fact_341_curryE,axiom,
    ! [F: fun_Pr1927655017t_bool,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(produc2091540094t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1555980039t_bool(F,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),Ba))) ) ).

tff(fact_342_curryE,axiom,
    ! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(hAPP_i119638232t_bool(produc97574839t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P178748804t_bool(F,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Aa),Ba))) ) ).

tff(fact_343_curryE,axiom,
    ! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(produc262399358t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P603027463t_bool(F,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba))) ) ).

tff(fact_344_one__reorient,axiom,
    ! [X_1: rat] :
      ( ( one_one_rat = X_1 )
    <=> ( X_1 = one_one_rat ) ) ).

tff(fact_345_one__reorient,axiom,
    ! [X_1: quickcheck_code_int] :
      ( ( one_on1684967323de_int = X_1 )
    <=> ( X_1 = one_on1684967323de_int ) ) ).

tff(fact_346_one__reorient,axiom,
    ! [X_1: complex] :
      ( ( one_one_complex = X_1 )
    <=> ( X_1 = one_one_complex ) ) ).

tff(fact_347_one__reorient,axiom,
    ! [X_1: code_code_numeral] :
      ( ( one_on1645066479umeral = X_1 )
    <=> ( X_1 = one_on1645066479umeral ) ) ).

tff(fact_348_one__reorient,axiom,
    ! [X_1: nat] :
      ( ( one_one_nat = X_1 )
    <=> ( X_1 = one_one_nat ) ) ).

tff(fact_349_one__reorient,axiom,
    ! [X_1: real] :
      ( ( one_one_real = X_1 )
    <=> ( X_1 = one_one_real ) ) ).

tff(fact_350_one__reorient,axiom,
    ! [X_1: int] :
      ( ( one_one_int = X_1 )
    <=> ( X_1 = one_one_int ) ) ).

tff(fact_351_zle__add1__eq__le,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ).

tff(fact_352_number__of__reorient,axiom,
    ! [W_1: int,X_1: rat] :
      ( ( hAPP_int_rat(number_number_of_rat,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_rat(number_number_of_rat,W_1) ) ) ).

tff(fact_353_number__of__reorient,axiom,
    ! [W_1: int,X_1: quickcheck_code_int] :
      ( ( hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_1 )
    <=> ( X_1 = hAPP_i1732201573de_int(number1226105091de_int,W_1) ) ) ).

tff(fact_354_number__of__reorient,axiom,
    ! [W_1: int,X_1: complex] :
      ( ( hAPP_int_complex(number528085621omplex,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_complex(number528085621omplex,W_1) ) ) ).

tff(fact_355_number__of__reorient,axiom,
    ! [W_1: int,X_1: code_code_numeral] :
      ( ( hAPP_i769753017umeral(number1443263063umeral,W_1) = X_1 )
    <=> ( X_1 = hAPP_i769753017umeral(number1443263063umeral,W_1) ) ) ).

tff(fact_356_number__of__reorient,axiom,
    ! [W_1: int,X_1: nat] :
      ( ( hAPP_int_nat(number_number_of_nat,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_nat(number_number_of_nat,W_1) ) ) ).

tff(fact_357_number__of__reorient,axiom,
    ! [W_1: int,X_1: real] :
      ( ( hAPP_int_real(number267125858f_real,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_real(number267125858f_real,W_1) ) ) ).

tff(fact_358_number__of__reorient,axiom,
    ! [W_1: int,X_1: int] :
      ( ( hAPP_int_int(number_number_of_int,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_int(number_number_of_int,W_1) ) ) ).

tff(fact_359_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,V)),hAPP_int_rat(number_number_of_rat,W_1)))
    <=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,W_1)),hAPP_int_rat(number_number_of_rat,V))) ) ).

tff(fact_360_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,V)),hAPP_i1732201573de_int(number1226105091de_int,W_1)))
    <=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,W_1)),hAPP_i1732201573de_int(number1226105091de_int,V))) ) ).

tff(fact_361_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,W_1)))
    <=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,W_1)),hAPP_int_nat(number_number_of_nat,V))) ) ).

tff(fact_362_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,V)),hAPP_int_real(number267125858f_real,W_1)))
    <=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,W_1)),hAPP_int_real(number267125858f_real,V))) ) ).

tff(fact_363_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1)))
    <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W_1)),hAPP_int_int(number_number_of_int,V))) ) ).

tff(fact_364_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,V)),hAPP_i769753017umeral(number1443263063umeral,W_1)))
    <=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,W_1)),hAPP_i769753017umeral(number1443263063umeral,V))) ) ).

tff(fact_365_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_rat(number_number_of_rat,X_1) = hAPP_int_rat(number_number_of_rat,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_366_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_complex(number528085621omplex,X_1) = hAPP_int_complex(number528085621omplex,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_367_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_real(number267125858f_real,X_1) = hAPP_int_real(number267125858f_real,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_368_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_int(number_number_of_int,X_1) = hAPP_int_int(number_number_of_int,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_369_less__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_370_less__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_371_less__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_372_le__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_373_le__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_374_le__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_375_add1__zle__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

tff(fact_376_linorder__neqE__linordered__idom,axiom,
    ! [X_70: rat,Y_56: rat] :
      ( ( X_70 != Y_56 )
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_70),Y_56))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_56),X_70)) ) ) ).

tff(fact_377_linorder__neqE__linordered__idom,axiom,
    ! [X_70: real,Y_56: real] :
      ( ( X_70 != Y_56 )
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_70),Y_56))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_56),X_70)) ) ) ).

tff(fact_378_linorder__neqE__linordered__idom,axiom,
    ! [X_70: int,Y_56: int] :
      ( ( X_70 != Y_56 )
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_70),Y_56))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_56),X_70)) ) ) ).

tff(fact_379_zless__imp__add1__zle,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1)) ) ).

tff(fact_380_order__le__neq__implies__less,axiom,
    ! [X_69: rat,Y_55: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_69),Y_55))
     => ( ( X_69 != Y_55 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_69),Y_55)) ) ) ).

tff(fact_381_order__le__neq__implies__less,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => ( ( X_1 != Y_2 )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2)) ) ) ).

tff(fact_382_order__le__neq__implies__less,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => ( ( X_1 != Y_2 )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2)) ) ) ).

tff(fact_383_order__le__neq__implies__less,axiom,
    ! [X_69: quickcheck_code_int,Y_55: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_69),Y_55))
     => ( ( X_69 != Y_55 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_69),Y_55)) ) ) ).

tff(fact_384_order__le__neq__implies__less,axiom,
    ! [X_69: nat,Y_55: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_69),Y_55))
     => ( ( X_69 != Y_55 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_69),Y_55)) ) ) ).

tff(fact_385_order__le__neq__implies__less,axiom,
    ! [X_69: real,Y_55: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_69),Y_55))
     => ( ( X_69 != Y_55 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_69),Y_55)) ) ) ).

tff(fact_386_order__le__neq__implies__less,axiom,
    ! [X_69: int,Y_55: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_69),Y_55))
     => ( ( X_69 != Y_55 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_69),Y_55)) ) ) ).

tff(fact_387_zle__diff1__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

tff(fact_388_less__1__mult,axiom,
    ! [N_8: rat,M_10: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_10))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_8))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_10),N_8))) ) ) ).

tff(fact_389_less__1__mult,axiom,
    ! [N_8: quickcheck_code_int,M_10: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_10))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_8))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_10),N_8))) ) ) ).

tff(fact_390_less__1__mult,axiom,
    ! [N_8: code_code_numeral,M_10: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_10))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_8))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_10),N_8))) ) ) ).

tff(fact_391_less__1__mult,axiom,
    ! [N_8: nat,M_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_10))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_8))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_10),N_8))) ) ) ).

tff(fact_392_less__1__mult,axiom,
    ! [N_8: real,M_10: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_10))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_8))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_10),N_8))) ) ) ).

tff(fact_393_less__1__mult,axiom,
    ! [N_8: int,M_10: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_10))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_8))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_10),N_8))) ) ) ).

tff(fact_394_less__add__one,axiom,
    ! [A_220: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_220),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_220),one_one_rat))) ).

tff(fact_395_less__add__one,axiom,
    ! [A_220: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_220),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_220),one_on1684967323de_int))) ).

tff(fact_396_less__add__one,axiom,
    ! [A_220: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_220),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_220),one_on1645066479umeral))) ).

tff(fact_397_less__add__one,axiom,
    ! [A_220: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_220),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_220),one_one_nat))) ).

tff(fact_398_less__add__one,axiom,
    ! [A_220: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_220),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_220),one_one_real))) ).

tff(fact_399_less__add__one,axiom,
    ! [A_220: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_220),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_220),one_one_int))) ).

tff(fact_400_add__le__less__mono,axiom,
    ! [C_85: rat,D_17: rat,A_219: rat,B_168: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_219),B_168))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_85),D_17))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_219),C_85)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_168),D_17))) ) ) ).

tff(fact_401_add__le__less__mono,axiom,
    ! [C_85: quickcheck_code_int,D_17: quickcheck_code_int,A_219: quickcheck_code_int,B_168: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_219),B_168))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_85),D_17))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_219),C_85)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_168),D_17))) ) ) ).

tff(fact_402_add__le__less__mono,axiom,
    ! [C_85: nat,D_17: nat,A_219: nat,B_168: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_219),B_168))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_85),D_17))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_219),C_85)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_168),D_17))) ) ) ).

tff(fact_403_add__le__less__mono,axiom,
    ! [C_85: real,D_17: real,A_219: real,B_168: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_219),B_168))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_85),D_17))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_219),C_85)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_168),D_17))) ) ) ).

tff(fact_404_add__le__less__mono,axiom,
    ! [C_85: int,D_17: int,A_219: int,B_168: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_219),B_168))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_85),D_17))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_219),C_85)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_168),D_17))) ) ) ).

tff(fact_405_add__less__le__mono,axiom,
    ! [C_84: rat,D_16: rat,A_218: rat,B_167: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_218),B_167))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_84),D_16))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_218),C_84)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_167),D_16))) ) ) ).

tff(fact_406_add__less__le__mono,axiom,
    ! [C_84: quickcheck_code_int,D_16: quickcheck_code_int,A_218: quickcheck_code_int,B_167: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_218),B_167))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_84),D_16))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_218),C_84)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_167),D_16))) ) ) ).

tff(fact_407_add__less__le__mono,axiom,
    ! [C_84: nat,D_16: nat,A_218: nat,B_167: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_218),B_167))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_84),D_16))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_218),C_84)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_167),D_16))) ) ) ).

tff(fact_408_add__less__le__mono,axiom,
    ! [C_84: real,D_16: real,A_218: real,B_167: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_218),B_167))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_84),D_16))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_218),C_84)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_167),D_16))) ) ) ).

tff(fact_409_add__less__le__mono,axiom,
    ! [C_84: int,D_16: int,A_218: int,B_167: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_218),B_167))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_84),D_16))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_218),C_84)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_167),D_16))) ) ) ).

tff(fact_410_zadd__zless__mono,axiom,
    ! [Z_4: int,Z_1: int,W_16: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_16),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_16),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1))) ) ) ).

tff(fact_411_zless__add1__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
        | ( W_1 = Z_2 ) ) ) ).

tff(fact_412_add__less__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Ba),C)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba)) ) ).

tff(fact_413_add__less__cancel__right,axiom,
    ! [Aa: quickcheck_code_int,C: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Aa),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Ba),C)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Aa),Ba)) ) ).

tff(fact_414_add__less__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),C)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).

tff(fact_415_add__less__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Ba),C)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).

tff(fact_416_add__less__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),C)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).

tff(fact_417_add__less__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),Ba)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba)) ) ).

tff(fact_418_add__less__cancel__left,axiom,
    ! [C: quickcheck_code_int,Aa: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),Aa)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),Ba)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Aa),Ba)) ) ).

tff(fact_419_add__less__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),Aa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),Ba)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).

tff(fact_420_add__less__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),Ba)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).

tff(fact_421_add__less__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),Ba)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).

tff(fact_422_add__strict__right__mono,axiom,
    ! [C_83: rat,A_217: rat,B_166: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_217),B_166))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_217),C_83)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_166),C_83))) ) ).

tff(fact_423_add__strict__right__mono,axiom,
    ! [C_83: quickcheck_code_int,A_217: quickcheck_code_int,B_166: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_217),B_166))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_217),C_83)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_166),C_83))) ) ).

tff(fact_424_add__strict__right__mono,axiom,
    ! [C_83: nat,A_217: nat,B_166: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_217),B_166))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_217),C_83)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_166),C_83))) ) ).

tff(fact_425_add__strict__right__mono,axiom,
    ! [C_83: real,A_217: real,B_166: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_217),B_166))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_217),C_83)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_166),C_83))) ) ).

tff(fact_426_add__strict__right__mono,axiom,
    ! [C_83: int,A_217: int,B_166: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_217),B_166))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_217),C_83)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_166),C_83))) ) ).

tff(fact_427_add__strict__left__mono,axiom,
    ! [C_82: rat,A_216: rat,B_165: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_216),B_165))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_82),A_216)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_82),B_165))) ) ).

tff(fact_428_add__strict__left__mono,axiom,
    ! [C_82: quickcheck_code_int,A_216: quickcheck_code_int,B_165: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_216),B_165))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_82),A_216)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_82),B_165))) ) ).

tff(fact_429_add__strict__left__mono,axiom,
    ! [C_82: nat,A_216: nat,B_165: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_216),B_165))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_82),A_216)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_82),B_165))) ) ).

tff(fact_430_add__strict__left__mono,axiom,
    ! [C_82: real,A_216: real,B_165: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_216),B_165))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_82),A_216)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_82),B_165))) ) ).

tff(fact_431_add__strict__left__mono,axiom,
    ! [C_82: int,A_216: int,B_165: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_216),B_165))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_82),A_216)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_82),B_165))) ) ).

tff(fact_432_add__strict__mono,axiom,
    ! [C_81: rat,D_15: rat,A_215: rat,B_164: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_215),B_164))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_81),D_15))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_215),C_81)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_164),D_15))) ) ) ).

tff(fact_433_add__strict__mono,axiom,
    ! [C_81: quickcheck_code_int,D_15: quickcheck_code_int,A_215: quickcheck_code_int,B_164: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_215),B_164))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_81),D_15))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_215),C_81)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_164),D_15))) ) ) ).

tff(fact_434_add__strict__mono,axiom,
    ! [C_81: nat,D_15: nat,A_215: nat,B_164: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_215),B_164))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_81),D_15))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_215),C_81)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_164),D_15))) ) ) ).

tff(fact_435_add__strict__mono,axiom,
    ! [C_81: real,D_15: real,A_215: real,B_164: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_215),B_164))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_81),D_15))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_215),C_81)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_164),D_15))) ) ) ).

tff(fact_436_add__strict__mono,axiom,
    ! [C_81: int,D_15: int,A_215: int,B_164: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_215),B_164))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_81),D_15))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_215),C_81)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_164),D_15))) ) ) ).

tff(fact_437_add__less__imp__less__right,axiom,
    ! [A_214: rat,C_80: rat,B_163: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_214),C_80)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_163),C_80)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_214),B_163)) ) ).

tff(fact_438_add__less__imp__less__right,axiom,
    ! [A_214: quickcheck_code_int,C_80: quickcheck_code_int,B_163: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_214),C_80)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_163),C_80)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_214),B_163)) ) ).

tff(fact_439_add__less__imp__less__right,axiom,
    ! [A_214: nat,C_80: nat,B_163: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_214),C_80)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_163),C_80)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_214),B_163)) ) ).

tff(fact_440_add__less__imp__less__right,axiom,
    ! [A_214: real,C_80: real,B_163: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_214),C_80)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_163),C_80)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_214),B_163)) ) ).

tff(fact_441_add__less__imp__less__right,axiom,
    ! [A_214: int,C_80: int,B_163: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_214),C_80)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_163),C_80)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_214),B_163)) ) ).

tff(fact_442_add__less__imp__less__left,axiom,
    ! [C_79: rat,A_213: rat,B_162: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_79),A_213)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_79),B_162)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_213),B_162)) ) ).

tff(fact_443_add__less__imp__less__left,axiom,
    ! [C_79: quickcheck_code_int,A_213: quickcheck_code_int,B_162: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_79),A_213)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_79),B_162)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_213),B_162)) ) ).

tff(fact_444_add__less__imp__less__left,axiom,
    ! [C_79: nat,A_213: nat,B_162: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_79),A_213)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_79),B_162)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_213),B_162)) ) ).

tff(fact_445_add__less__imp__less__left,axiom,
    ! [C_79: real,A_213: real,B_162: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_79),A_213)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_79),B_162)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_213),B_162)) ) ).

tff(fact_446_add__less__imp__less__left,axiom,
    ! [C_79: int,A_213: int,B_162: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_79),A_213)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_79),B_162)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_213),B_162)) ) ).

tff(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_212) = A_212 ).

tff(fact_448_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_212) = A_212 ).

tff(fact_449_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_212) = A_212 ).

tff(fact_450_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_212) = A_212 ).

tff(fact_451_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_212) = A_212 ).

tff(fact_452_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_212) = A_212 ).

tff(fact_453_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_212: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_212) = A_212 ).

tff(fact_454_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_211),one_one_rat) = A_211 ).

tff(fact_455_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_211),one_on1684967323de_int) = A_211 ).

tff(fact_456_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_211),one_one_complex) = A_211 ).

tff(fact_457_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_211),one_on1645066479umeral) = A_211 ).

tff(fact_458_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_211),one_one_nat) = A_211 ).

tff(fact_459_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_211),one_one_real) = A_211 ).

tff(fact_460_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_211: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_211),one_one_int) = A_211 ).

tff(fact_461_mult__1__left,axiom,
    ! [A_210: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_210) = A_210 ).

tff(fact_462_mult__1__left,axiom,
    ! [A_210: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_210) = A_210 ).

tff(fact_463_mult__1__left,axiom,
    ! [A_210: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_210) = A_210 ).

tff(fact_464_mult__1__left,axiom,
    ! [A_210: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_210) = A_210 ).

tff(fact_465_mult__1__left,axiom,
    ! [A_210: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_210) = A_210 ).

tff(fact_466_mult__1__left,axiom,
    ! [A_210: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_210) = A_210 ).

tff(fact_467_mult__1__left,axiom,
    ! [A_210: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_210) = A_210 ).

tff(fact_468_mult__1,axiom,
    ! [A_209: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_209) = A_209 ).

tff(fact_469_mult__1,axiom,
    ! [A_209: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_209) = A_209 ).

tff(fact_470_mult__1,axiom,
    ! [A_209: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_209) = A_209 ).

tff(fact_471_mult__1,axiom,
    ! [A_209: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_209) = A_209 ).

tff(fact_472_mult__1,axiom,
    ! [A_209: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_209) = A_209 ).

tff(fact_473_mult__1,axiom,
    ! [A_209: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_209) = A_209 ).

tff(fact_474_mult__1,axiom,
    ! [A_209: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_209) = A_209 ).

tff(fact_475_mult__1__right,axiom,
    ! [A_208: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_208),one_one_rat) = A_208 ).

tff(fact_476_mult__1__right,axiom,
    ! [A_208: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_208),one_on1684967323de_int) = A_208 ).

tff(fact_477_mult__1__right,axiom,
    ! [A_208: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_208),one_one_complex) = A_208 ).

tff(fact_478_mult__1__right,axiom,
    ! [A_208: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_208),one_on1645066479umeral) = A_208 ).

tff(fact_479_mult__1__right,axiom,
    ! [A_208: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_208),one_one_nat) = A_208 ).

tff(fact_480_mult__1__right,axiom,
    ! [A_208: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_208),one_one_real) = A_208 ).

tff(fact_481_mult__1__right,axiom,
    ! [A_208: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_208),one_one_int) = A_208 ).

tff(fact_482_mult_Ocomm__neutral,axiom,
    ! [A_207: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_207),one_one_rat) = A_207 ).

tff(fact_483_mult_Ocomm__neutral,axiom,
    ! [A_207: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_207),one_on1684967323de_int) = A_207 ).

tff(fact_484_mult_Ocomm__neutral,axiom,
    ! [A_207: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_207),one_one_complex) = A_207 ).

tff(fact_485_mult_Ocomm__neutral,axiom,
    ! [A_207: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_207),one_on1645066479umeral) = A_207 ).

tff(fact_486_mult_Ocomm__neutral,axiom,
    ! [A_207: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_207),one_one_nat) = A_207 ).

tff(fact_487_mult_Ocomm__neutral,axiom,
    ! [A_207: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_207),one_one_real) = A_207 ).

tff(fact_488_mult_Ocomm__neutral,axiom,
    ! [A_207: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_207),one_one_int) = A_207 ).

tff(fact_489_diff__eq__diff__less,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D)) ) ) ).

tff(fact_490_diff__eq__diff__less,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D)) ) ) ).

tff(fact_491_diff__eq__diff__less,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D)) ) ) ).

tff(fact_492_add__le__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Ba),C)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba)) ) ).

tff(fact_493_add__le__cancel__right,axiom,
    ! [Aa: quickcheck_code_int,C: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Aa),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Ba),C)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Aa),Ba)) ) ).

tff(fact_494_add__le__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),C)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).

tff(fact_495_add__le__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Ba),C)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).

tff(fact_496_add__le__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),C)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).

tff(fact_497_add__le__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),Ba)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba)) ) ).

tff(fact_498_add__le__cancel__left,axiom,
    ! [C: quickcheck_code_int,Aa: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),Aa)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),Ba)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Aa),Ba)) ) ).

tff(fact_499_add__le__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),Aa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),Ba)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).

tff(fact_500_add__le__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),Ba)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).

tff(fact_501_add__le__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),Ba)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).

tff(fact_502_add__right__mono,axiom,
    ! [C_78: rat,A_206: rat,B_161: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_206),B_161))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_206),C_78)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_161),C_78))) ) ).

tff(fact_503_add__right__mono,axiom,
    ! [C_78: quickcheck_code_int,A_206: quickcheck_code_int,B_161: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_206),B_161))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_206),C_78)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_161),C_78))) ) ).

tff(fact_504_add__right__mono,axiom,
    ! [C_78: nat,A_206: nat,B_161: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_206),B_161))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_206),C_78)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_161),C_78))) ) ).

tff(fact_505_add__right__mono,axiom,
    ! [C_78: real,A_206: real,B_161: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_206),B_161))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_206),C_78)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_161),C_78))) ) ).

tff(fact_506_add__right__mono,axiom,
    ! [C_78: int,A_206: int,B_161: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_206),B_161))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_206),C_78)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_161),C_78))) ) ).

tff(fact_507_add__left__mono,axiom,
    ! [C_77: rat,A_205: rat,B_160: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_205),B_160))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_77),A_205)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_77),B_160))) ) ).

tff(fact_508_add__left__mono,axiom,
    ! [C_77: quickcheck_code_int,A_205: quickcheck_code_int,B_160: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_205),B_160))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_77),A_205)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_77),B_160))) ) ).

tff(fact_509_add__left__mono,axiom,
    ! [C_77: nat,A_205: nat,B_160: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_205),B_160))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_77),A_205)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_77),B_160))) ) ).

tff(fact_510_add__left__mono,axiom,
    ! [C_77: real,A_205: real,B_160: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_205),B_160))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_77),A_205)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_77),B_160))) ) ).

tff(fact_511_add__left__mono,axiom,
    ! [C_77: int,A_205: int,B_160: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_205),B_160))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_77),A_205)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_77),B_160))) ) ).

tff(fact_512_add__mono,axiom,
    ! [C_76: rat,D_14: rat,A_204: rat,B_159: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_204),B_159))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_76),D_14))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_204),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_159),D_14))) ) ) ).

tff(fact_513_add__mono,axiom,
    ! [C_76: quickcheck_code_int,D_14: quickcheck_code_int,A_204: quickcheck_code_int,B_159: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_204),B_159))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_76),D_14))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_204),C_76)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_159),D_14))) ) ) ).

tff(fact_514_add__mono,axiom,
    ! [C_76: nat,D_14: nat,A_204: nat,B_159: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_204),B_159))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_76),D_14))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_204),C_76)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_159),D_14))) ) ) ).

tff(fact_515_add__mono,axiom,
    ! [C_76: real,D_14: real,A_204: real,B_159: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_204),B_159))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_76),D_14))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_204),C_76)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_159),D_14))) ) ) ).

tff(fact_516_add__mono,axiom,
    ! [C_76: int,D_14: int,A_204: int,B_159: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_204),B_159))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_76),D_14))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_204),C_76)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_159),D_14))) ) ) ).

tff(fact_517_add__le__imp__le__right,axiom,
    ! [A_203: rat,C_75: rat,B_158: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_203),C_75)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_158),C_75)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_203),B_158)) ) ).

tff(fact_518_add__le__imp__le__right,axiom,
    ! [A_203: quickcheck_code_int,C_75: quickcheck_code_int,B_158: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_203),C_75)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_158),C_75)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_203),B_158)) ) ).

tff(fact_519_add__le__imp__le__right,axiom,
    ! [A_203: nat,C_75: nat,B_158: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_203),C_75)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_158),C_75)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_203),B_158)) ) ).

tff(fact_520_add__le__imp__le__right,axiom,
    ! [A_203: real,C_75: real,B_158: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_203),C_75)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_158),C_75)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_203),B_158)) ) ).

tff(fact_521_add__le__imp__le__right,axiom,
    ! [A_203: int,C_75: int,B_158: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_203),C_75)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_158),C_75)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_203),B_158)) ) ).

tff(fact_522_add__le__imp__le__left,axiom,
    ! [C_74: rat,A_202: rat,B_157: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_74),A_202)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_74),B_157)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_202),B_157)) ) ).

tff(fact_523_add__le__imp__le__left,axiom,
    ! [C_74: quickcheck_code_int,A_202: quickcheck_code_int,B_157: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_74),A_202)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_74),B_157)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_202),B_157)) ) ).

tff(fact_524_add__le__imp__le__left,axiom,
    ! [C_74: nat,A_202: nat,B_157: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_74),A_202)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_74),B_157)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_202),B_157)) ) ).

tff(fact_525_add__le__imp__le__left,axiom,
    ! [C_74: real,A_202: real,B_157: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_74),A_202)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_74),B_157)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_202),B_157)) ) ).

tff(fact_526_add__le__imp__le__left,axiom,
    ! [C_74: int,A_202: int,B_157: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_74),A_202)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_74),B_157)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_202),B_157)) ) ).

tff(fact_527_diff__eq__diff__less__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D)) ) ) ).

tff(fact_528_diff__eq__diff__less__eq,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D)) ) ) ).

tff(fact_529_diff__eq__diff__less__eq,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D)) ) ) ).

tff(fact_530_times__numeral__code_I5_J,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),W)) ).

tff(fact_531_plus__numeral__code_I9_J,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),W)) ).

tff(fact_532_zadd__strict__right__mono,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J_2),K_2))) ) ).

tff(fact_533_zadd__left__mono,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),J_2))) ) ).

tff(fact_534_zmult__1,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z_1) = Z_1 ).

tff(fact_535_zmult__1__right,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),one_one_int) = Z_1 ).

tff(fact_536_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: rat,C_73: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_156),C_73)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_24)),B_156)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_24)),C_73)) ).

tff(fact_537_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: quickcheck_code_int,C_73: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_156),C_73)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_24)),B_156)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_24)),C_73)) ).

tff(fact_538_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: complex,C_73: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_24)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_156),C_73)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_24)),B_156)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_24)),C_73)) ).

tff(fact_539_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: nat,C_73: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_156),C_73)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_24)),B_156)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_24)),C_73)) ).

tff(fact_540_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: real,C_73: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_24)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_156),C_73)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_24)),B_156)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_24)),C_73)) ).

tff(fact_541_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: int,C_73: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_24)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_156),C_73)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_24)),B_156)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_24)),C_73)) ).

tff(fact_542_right__distrib__number__of,axiom,
    ! [V_24: int,B_156: code_code_numeral,C_73: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_156),C_73)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_24)),B_156)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_24)),C_73)) ).

tff(fact_543_left__distrib__number__of,axiom,
    ! [A_201: rat,B_155: rat,V_23: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_201),B_155)),hAPP_int_rat(number_number_of_rat,V_23)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_201),hAPP_int_rat(number_number_of_rat,V_23))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_155),hAPP_int_rat(number_number_of_rat,V_23))) ).

tff(fact_544_left__distrib__number__of,axiom,
    ! [A_201: quickcheck_code_int,B_155: quickcheck_code_int,V_23: int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_201),B_155)),hAPP_i1732201573de_int(number1226105091de_int,V_23)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_201),hAPP_i1732201573de_int(number1226105091de_int,V_23))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_155),hAPP_i1732201573de_int(number1226105091de_int,V_23))) ).

tff(fact_545_left__distrib__number__of,axiom,
    ! [A_201: complex,B_155: complex,V_23: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_201),B_155)),hAPP_int_complex(number528085621omplex,V_23)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_201),hAPP_int_complex(number528085621omplex,V_23))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_155),hAPP_int_complex(number528085621omplex,V_23))) ).

tff(fact_546_left__distrib__number__of,axiom,
    ! [A_201: nat,B_155: nat,V_23: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_201),B_155)),hAPP_int_nat(number_number_of_nat,V_23)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_201),hAPP_int_nat(number_number_of_nat,V_23))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_155),hAPP_int_nat(number_number_of_nat,V_23))) ).

tff(fact_547_left__distrib__number__of,axiom,
    ! [A_201: real,B_155: real,V_23: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_201),B_155)),hAPP_int_real(number267125858f_real,V_23)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_201),hAPP_int_real(number267125858f_real,V_23))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_155),hAPP_int_real(number267125858f_real,V_23))) ).

tff(fact_548_left__distrib__number__of,axiom,
    ! [A_201: int,B_155: int,V_23: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_201),B_155)),hAPP_int_int(number_number_of_int,V_23)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_201),hAPP_int_int(number_number_of_int,V_23))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_155),hAPP_int_int(number_number_of_int,V_23))) ).

tff(fact_549_left__distrib__number__of,axiom,
    ! [A_201: code_code_numeral,B_155: code_code_numeral,V_23: int] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_201),B_155)),hAPP_i769753017umeral(number1443263063umeral,V_23)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_201),hAPP_i769753017umeral(number1443263063umeral,V_23))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_155),hAPP_i769753017umeral(number1443263063umeral,V_23))) ).

tff(fact_550_right__diff__distrib__number__of,axiom,
    ! [V_22: int,B_154: rat,C_72: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_22)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_154),C_72)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_22)),B_154)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_22)),C_72)) ).

tff(fact_551_right__diff__distrib__number__of,axiom,
    ! [V_22: int,B_154: complex,C_72: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_22)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_154),C_72)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_22)),B_154)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_22)),C_72)) ).

tff(fact_552_right__diff__distrib__number__of,axiom,
    ! [V_22: int,B_154: real,C_72: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_22)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_154),C_72)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_22)),B_154)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_22)),C_72)) ).

tff(fact_553_right__diff__distrib__number__of,axiom,
    ! [V_22: int,B_154: int,C_72: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_22)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_154),C_72)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_22)),B_154)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_22)),C_72)) ).

tff(fact_554_left__diff__distrib__number__of,axiom,
    ! [A_200: rat,B_153: rat,V_21: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_200),B_153)),hAPP_int_rat(number_number_of_rat,V_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_200),hAPP_int_rat(number_number_of_rat,V_21))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_153),hAPP_int_rat(number_number_of_rat,V_21))) ).

tff(fact_555_left__diff__distrib__number__of,axiom,
    ! [A_200: complex,B_153: complex,V_21: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_200),B_153)),hAPP_int_complex(number528085621omplex,V_21)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_200),hAPP_int_complex(number528085621omplex,V_21))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_153),hAPP_int_complex(number528085621omplex,V_21))) ).

tff(fact_556_left__diff__distrib__number__of,axiom,
    ! [A_200: real,B_153: real,V_21: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_200),B_153)),hAPP_int_real(number267125858f_real,V_21)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_200),hAPP_int_real(number267125858f_real,V_21))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_153),hAPP_int_real(number267125858f_real,V_21))) ).

tff(fact_557_left__diff__distrib__number__of,axiom,
    ! [A_200: int,B_153: int,V_21: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_200),B_153)),hAPP_int_int(number_number_of_int,V_21)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_200),hAPP_int_int(number_number_of_int,V_21))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_153),hAPP_int_int(number_number_of_int,V_21))) ).

tff(fact_558_number__of__mult,axiom,
    ! [V_20: int,W_15: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_20)),hAPP_int_rat(number_number_of_rat,W_15)) ).

tff(fact_559_number__of__mult,axiom,
    ! [V_20: int,W_15: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_15)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_20)),hAPP_int_complex(number528085621omplex,W_15)) ).

tff(fact_560_number__of__mult,axiom,
    ! [V_20: int,W_15: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_15)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_20)),hAPP_int_real(number267125858f_real,W_15)) ).

tff(fact_561_number__of__mult,axiom,
    ! [V_20: int,W_15: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_15)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_20)),hAPP_int_int(number_number_of_int,W_15)) ).

tff(fact_562_arith__simps_I32_J,axiom,
    ! [V_19: int,W_14: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_19)),hAPP_int_rat(number_number_of_rat,W_14)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_14)) ).

tff(fact_563_arith__simps_I32_J,axiom,
    ! [V_19: int,W_14: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_19)),hAPP_int_complex(number528085621omplex,W_14)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_14)) ).

tff(fact_564_arith__simps_I32_J,axiom,
    ! [V_19: int,W_14: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_19)),hAPP_int_real(number267125858f_real,W_14)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_14)) ).

tff(fact_565_arith__simps_I32_J,axiom,
    ! [V_19: int,W_14: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_19)),hAPP_int_int(number_number_of_int,W_14)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_14)) ).

tff(fact_566_mult__number__of__left,axiom,
    ! [V_18: int,W_13: int,Z_21: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_18)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_13)),Z_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_13))),Z_21) ).

tff(fact_567_mult__number__of__left,axiom,
    ! [V_18: int,W_13: int,Z_21: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_18)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_13)),Z_21)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_13))),Z_21) ).

tff(fact_568_mult__number__of__left,axiom,
    ! [V_18: int,W_13: int,Z_21: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_18)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_13)),Z_21)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_13))),Z_21) ).

tff(fact_569_mult__number__of__left,axiom,
    ! [V_18: int,W_13: int,Z_21: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_18)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_13)),Z_21)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_13))),Z_21) ).

tff(fact_570_number__of__add,axiom,
    ! [V_17: int,W_12: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_17),W_12)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_17)),hAPP_int_rat(number_number_of_rat,W_12)) ).

tff(fact_571_number__of__add,axiom,
    ! [V_17: int,W_12: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_17),W_12)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_17)),hAPP_int_complex(number528085621omplex,W_12)) ).

tff(fact_572_number__of__add,axiom,
    ! [V_17: int,W_12: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_17),W_12)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_17)),hAPP_int_real(number267125858f_real,W_12)) ).

tff(fact_573_number__of__add,axiom,
    ! [V_17: int,W_12: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_17),W_12)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_17)),hAPP_int_int(number_number_of_int,W_12)) ).

tff(fact_574_add__number__of__eq,axiom,
    ! [V_16: int,W_11: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_16)),hAPP_int_rat(number_number_of_rat,W_11)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),W_11)) ).

tff(fact_575_add__number__of__eq,axiom,
    ! [V_16: int,W_11: int] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_16)),hAPP_int_complex(number528085621omplex,W_11)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),W_11)) ).

tff(fact_576_add__number__of__eq,axiom,
    ! [V_16: int,W_11: int] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_16)),hAPP_int_real(number267125858f_real,W_11)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),W_11)) ).

tff(fact_577_add__number__of__eq,axiom,
    ! [V_16: int,W_11: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_16)),hAPP_int_int(number_number_of_int,W_11)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),W_11)) ).

tff(fact_578_add__number__of__left,axiom,
    ! [V_15: int,W_10: int,Z_20: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_15)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_10)),Z_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_15),W_10))),Z_20) ).

tff(fact_579_add__number__of__left,axiom,
    ! [V_15: int,W_10: int,Z_20: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_15)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_10)),Z_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_15),W_10))),Z_20) ).

tff(fact_580_add__number__of__left,axiom,
    ! [V_15: int,W_10: int,Z_20: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_15)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_10)),Z_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_15),W_10))),Z_20) ).

tff(fact_581_add__number__of__left,axiom,
    ! [V_15: int,W_10: int,Z_20: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_15)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_10)),Z_20)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_15),W_10))),Z_20) ).

tff(fact_582_number__of__diff,axiom,
    ! [V_14: int,W_9: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_14),W_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_14)),hAPP_int_rat(number_number_of_rat,W_9)) ).

tff(fact_583_number__of__diff,axiom,
    ! [V_14: int,W_9: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_14),W_9)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_int_complex(number528085621omplex,W_9)) ).

tff(fact_584_number__of__diff,axiom,
    ! [V_14: int,W_9: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_14),W_9)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_int_real(number267125858f_real,W_9)) ).

tff(fact_585_number__of__diff,axiom,
    ! [V_14: int,W_9: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_14),W_9)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_14)),hAPP_int_int(number_number_of_int,W_9)) ).

tff(fact_586_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_9),M_9) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),M_9) ).

tff(fact_587_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_9),M_9) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int)),M_9) ).

tff(fact_588_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_9),M_9) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_9) ).

tff(fact_589_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_9),M_9) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_9) ).

tff(fact_590_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_9),M_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_9) ).

tff(fact_591_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_9),M_9) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),M_9) ).

tff(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_9: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_9),M_9) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_9) ).

tff(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: rat,A_199: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_8),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_199),M_8)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_199),one_one_rat)),M_8) ).

tff(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: quickcheck_code_int,A_199: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_8),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_199),M_8)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_199),one_on1684967323de_int)),M_8) ).

tff(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: complex,A_199: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_8),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_199),M_8)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_199),one_one_complex)),M_8) ).

tff(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: code_code_numeral,A_199: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_8),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_199),M_8)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_199),one_on1645066479umeral)),M_8) ).

tff(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: nat,A_199: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_199),M_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_199),one_one_nat)),M_8) ).

tff(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: real,A_199: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_8),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_199),M_8)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_199),one_one_real)),M_8) ).

tff(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_8: int,A_199: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_8),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_199),M_8)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_199),one_one_int)),M_8) ).

tff(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: rat,M_7: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_198),M_7)),M_7) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_198),one_one_rat)),M_7) ).

tff(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: quickcheck_code_int,M_7: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_198),M_7)),M_7) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_198),one_on1684967323de_int)),M_7) ).

tff(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: complex,M_7: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_198),M_7)),M_7) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_198),one_one_complex)),M_7) ).

tff(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: code_code_numeral,M_7: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_198),M_7)),M_7) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_198),one_on1645066479umeral)),M_7) ).

tff(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: nat,M_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_198),M_7)),M_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_198),one_one_nat)),M_7) ).

tff(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: real,M_7: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_198),M_7)),M_7) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_198),one_one_real)),M_7) ).

tff(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_198: int,M_7: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_198),M_7)),M_7) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_198),one_one_int)),M_7) ).

tff(fact_607_curryD,axiom,
    ! [F: fun_Pr4007113l_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(produc1775888740l_bool(F),Aa),Ba))
     => hBOOL(hAPP_P880235623l_bool(F,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,Aa),Ba))) ) ).

tff(fact_608_curryD,axiom,
    ! [F: fun_Pr1333588703t_bool,Aa: product_prod_int_int,Ba: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(hAPP_P989584703t_bool(produc353512698t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P62333565t_bool(F,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,Aa),Ba))) ) ).

tff(fact_609_curryD,axiom,
    ! [F: fun_Pr610342249t_bool,Aa: quickcheck_code_int,Ba: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(produc2013771390t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1709355527t_bool(F,hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,Aa),Ba))) ) ).

tff(fact_610_curryD,axiom,
    ! [F: fun_Pr854501993l_bool,Aa: code_code_numeral,Ba: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(produc609372286l_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1164677895l_bool(F,hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,Aa),Ba))) ) ).

tff(fact_611_curryD,axiom,
    ! [F: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(produc748992778l_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1333854989l_bool(F,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba))) ) ).

tff(fact_612_curryD,axiom,
    ! [F: fun_Pr587612383t_bool,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(hAPP_P705731903t_bool(produc1319041786t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1463840893t_bool(F,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,Aa),Ba))) ) ).

tff(fact_613_curryD,axiom,
    ! [F: fun_Pr1927655017t_bool,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(produc2091540094t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P1555980039t_bool(F,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),Ba))) ) ).

tff(fact_614_curryD,axiom,
    ! [F: fun_Pr1503267378t_bool,Aa: int,Ba: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(hAPP_i119638232t_bool(produc97574839t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P178748804t_bool(F,hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Aa),Ba))) ) ).

tff(fact_615_curryD,axiom,
    ! [F: fun_Pr974702441t_bool,Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(produc262399358t_bool(F),Aa),Ba))
     => hBOOL(hAPP_P603027463t_bool(F,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba))) ) ).

tff(fact_616_order__refl,axiom,
    ! [X_68: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_68),X_68)) ).

tff(fact_617_order__refl,axiom,
    ! [X_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),X_1)) ).

tff(fact_618_order__refl,axiom,
    ! [X_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),X_1)) ).

tff(fact_619_order__refl,axiom,
    ! [X_68: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_68),X_68)) ).

tff(fact_620_order__refl,axiom,
    ! [X_68: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_68),X_68)) ).

tff(fact_621_order__refl,axiom,
    ! [X_68: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_68),X_68)) ).

tff(fact_622_order__refl,axiom,
    ! [X_68: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_68),X_68)) ).

tff(fact_623_xt1_I8_J,axiom,
    ! [Z_19: rat,Y_54: rat,X_67: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_54),X_67))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_19),Y_54))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_19),X_67)) ) ) ).

tff(fact_624_xt1_I8_J,axiom,
    ! [Z_2: fun_int_bool,Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),X_1)) ) ) ).

tff(fact_625_xt1_I8_J,axiom,
    ! [Z_2: fun_nat_bool,Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),X_1)) ) ) ).

tff(fact_626_xt1_I8_J,axiom,
    ! [Z_19: quickcheck_code_int,Y_54: quickcheck_code_int,X_67: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_54),X_67))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_19),Y_54))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_19),X_67)) ) ) ).

tff(fact_627_xt1_I8_J,axiom,
    ! [Z_19: nat,Y_54: nat,X_67: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_54),X_67))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_19),Y_54))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_19),X_67)) ) ) ).

tff(fact_628_xt1_I8_J,axiom,
    ! [Z_19: real,Y_54: real,X_67: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_54),X_67))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_19),Y_54))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_19),X_67)) ) ) ).

tff(fact_629_xt1_I8_J,axiom,
    ! [Z_19: int,Y_54: int,X_67: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_54),X_67))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_19),Y_54))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_19),X_67)) ) ) ).

tff(fact_630_order__le__less__trans,axiom,
    ! [Z_18: rat,X_66: rat,Y_53: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_66),Y_53))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_53),Z_18))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_66),Z_18)) ) ) ).

tff(fact_631_order__le__less__trans,axiom,
    ! [Z_2: fun_int_bool,X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Z_2)) ) ) ).

tff(fact_632_order__le__less__trans,axiom,
    ! [Z_2: fun_nat_bool,X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Z_2)) ) ) ).

tff(fact_633_order__le__less__trans,axiom,
    ! [Z_18: quickcheck_code_int,X_66: quickcheck_code_int,Y_53: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_66),Y_53))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_53),Z_18))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_66),Z_18)) ) ) ).

tff(fact_634_order__le__less__trans,axiom,
    ! [Z_18: nat,X_66: nat,Y_53: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_66),Y_53))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_53),Z_18))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_66),Z_18)) ) ) ).

tff(fact_635_order__le__less__trans,axiom,
    ! [Z_18: real,X_66: real,Y_53: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_66),Y_53))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_53),Z_18))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_66),Z_18)) ) ) ).

tff(fact_636_order__le__less__trans,axiom,
    ! [Z_18: int,X_66: int,Y_53: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_66),Y_53))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_53),Z_18))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_66),Z_18)) ) ) ).

tff(fact_637_xt1_I7_J,axiom,
    ! [Z_17: rat,Y_52: rat,X_65: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_52),X_65))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_17),Y_52))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_17),X_65)) ) ) ).

tff(fact_638_xt1_I7_J,axiom,
    ! [Z_2: fun_int_bool,Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),X_1)) ) ) ).

tff(fact_639_xt1_I7_J,axiom,
    ! [Z_2: fun_nat_bool,Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),X_1)) ) ) ).

tff(fact_640_xt1_I7_J,axiom,
    ! [Z_17: quickcheck_code_int,Y_52: quickcheck_code_int,X_65: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_52),X_65))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_17),Y_52))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_17),X_65)) ) ) ).

tff(fact_641_xt1_I7_J,axiom,
    ! [Z_17: nat,Y_52: nat,X_65: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_52),X_65))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_17),Y_52))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_17),X_65)) ) ) ).

tff(fact_642_xt1_I7_J,axiom,
    ! [Z_17: real,Y_52: real,X_65: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_52),X_65))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_17),Y_52))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_17),X_65)) ) ) ).

tff(fact_643_xt1_I7_J,axiom,
    ! [Z_17: int,Y_52: int,X_65: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_52),X_65))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_17),Y_52))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_17),X_65)) ) ) ).

tff(fact_644_order__less__le__trans,axiom,
    ! [Z_16: rat,X_64: rat,Y_51: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_64),Y_51))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_51),Z_16))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_64),Z_16)) ) ) ).

tff(fact_645_order__less__le__trans,axiom,
    ! [Z_2: fun_int_bool,X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Z_2)) ) ) ).

tff(fact_646_order__less__le__trans,axiom,
    ! [Z_2: fun_nat_bool,X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Z_2)) ) ) ).

tff(fact_647_order__less__le__trans,axiom,
    ! [Z_16: quickcheck_code_int,X_64: quickcheck_code_int,Y_51: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_64),Y_51))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_51),Z_16))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_64),Z_16)) ) ) ).

tff(fact_648_order__less__le__trans,axiom,
    ! [Z_16: nat,X_64: nat,Y_51: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_64),Y_51))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_51),Z_16))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_64),Z_16)) ) ) ).

tff(fact_649_order__less__le__trans,axiom,
    ! [Z_16: real,X_64: real,Y_51: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_64),Y_51))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_51),Z_16))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_64),Z_16)) ) ) ).

tff(fact_650_order__less__le__trans,axiom,
    ! [Z_16: int,X_64: int,Y_51: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_64),Y_51))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_51),Z_16))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_64),Z_16)) ) ) ).

tff(fact_651_xt1_I11_J,axiom,
    ! [B_152: rat,A_197: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_152),A_197))
     => ( ( A_197 != B_152 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_152),A_197)) ) ) ).

tff(fact_652_xt1_I11_J,axiom,
    ! [Ba: fun_int_bool,Aa: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),Aa))
     => ( ( Aa != Ba )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ba),Aa)) ) ) ).

tff(fact_653_xt1_I11_J,axiom,
    ! [Ba: fun_nat_bool,Aa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),Aa))
     => ( ( Aa != Ba )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa)) ) ) ).

tff(fact_654_xt1_I11_J,axiom,
    ! [B_152: quickcheck_code_int,A_197: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_152),A_197))
     => ( ( A_197 != B_152 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_152),A_197)) ) ) ).

tff(fact_655_xt1_I11_J,axiom,
    ! [B_152: nat,A_197: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_152),A_197))
     => ( ( A_197 != B_152 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_152),A_197)) ) ) ).

tff(fact_656_xt1_I11_J,axiom,
    ! [B_152: real,A_197: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_152),A_197))
     => ( ( A_197 != B_152 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_152),A_197)) ) ) ).

tff(fact_657_xt1_I11_J,axiom,
    ! [B_152: int,A_197: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_152),A_197))
     => ( ( A_197 != B_152 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_152),A_197)) ) ) ).

tff(fact_658_order__le__neq__trans,axiom,
    ! [A_196: rat,B_151: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_196),B_151))
     => ( ( A_196 != B_151 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_196),B_151)) ) ) ).

tff(fact_659_order__le__neq__trans,axiom,
    ! [Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Aa),Ba))
     => ( ( Aa != Ba )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),Ba)) ) ) ).

tff(fact_660_order__le__neq__trans,axiom,
    ! [Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Aa),Ba))
     => ( ( Aa != Ba )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba)) ) ) ).

tff(fact_661_order__le__neq__trans,axiom,
    ! [A_196: quickcheck_code_int,B_151: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_196),B_151))
     => ( ( A_196 != B_151 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_196),B_151)) ) ) ).

tff(fact_662_order__le__neq__trans,axiom,
    ! [A_196: nat,B_151: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_196),B_151))
     => ( ( A_196 != B_151 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_196),B_151)) ) ) ).

tff(fact_663_order__le__neq__trans,axiom,
    ! [A_196: real,B_151: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_196),B_151))
     => ( ( A_196 != B_151 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_196),B_151)) ) ) ).

tff(fact_664_order__le__neq__trans,axiom,
    ! [A_196: int,B_151: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_196),B_151))
     => ( ( A_196 != B_151 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_196),B_151)) ) ) ).

tff(fact_665_order__le__imp__less__or__eq,axiom,
    ! [X_63: rat,Y_50: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_63),Y_50))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_63),Y_50))
        | ( X_63 = Y_50 ) ) ) ).

tff(fact_666_order__le__imp__less__or__eq,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_667_order__le__imp__less__or__eq,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_668_order__le__imp__less__or__eq,axiom,
    ! [X_63: quickcheck_code_int,Y_50: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_63),Y_50))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_63),Y_50))
        | ( X_63 = Y_50 ) ) ) ).

tff(fact_669_order__le__imp__less__or__eq,axiom,
    ! [X_63: nat,Y_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_63),Y_50))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_63),Y_50))
        | ( X_63 = Y_50 ) ) ) ).

tff(fact_670_order__le__imp__less__or__eq,axiom,
    ! [X_63: real,Y_50: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_63),Y_50))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_63),Y_50))
        | ( X_63 = Y_50 ) ) ) ).

tff(fact_671_order__le__imp__less__or__eq,axiom,
    ! [X_63: int,Y_50: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_63),Y_50))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_63),Y_50))
        | ( X_63 = Y_50 ) ) ) ).

tff(fact_672_linorder__antisym__conv2,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_673_linorder__antisym__conv2,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
     => ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_674_linorder__antisym__conv2,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_675_linorder__antisym__conv2,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_676_linorder__antisym__conv2,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_677_order__less__imp__le,axiom,
    ! [X_62: rat,Y_49: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_62),Y_49))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_62),Y_49)) ) ).

tff(fact_678_order__less__imp__le,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2)) ) ).

tff(fact_679_order__less__imp__le,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2)) ) ).

tff(fact_680_order__less__imp__le,axiom,
    ! [X_62: quickcheck_code_int,Y_49: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_62),Y_49))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_62),Y_49)) ) ).

tff(fact_681_order__less__imp__le,axiom,
    ! [X_62: nat,Y_49: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_62),Y_49))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_62),Y_49)) ) ).

tff(fact_682_order__less__imp__le,axiom,
    ! [X_62: real,Y_49: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_62),Y_49))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_62),Y_49)) ) ).

tff(fact_683_order__less__imp__le,axiom,
    ! [X_62: int,Y_49: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_62),Y_49))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_62),Y_49)) ) ).

tff(fact_684_leD,axiom,
    ! [Y_48: rat,X_61: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_48),X_61))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_61),Y_48)) ) ).

tff(fact_685_leD,axiom,
    ! [Y_48: quickcheck_code_int,X_61: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_48),X_61))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_61),Y_48)) ) ).

tff(fact_686_leD,axiom,
    ! [Y_48: nat,X_61: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_48),X_61))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_61),Y_48)) ) ).

tff(fact_687_leD,axiom,
    ! [Y_48: real,X_61: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_48),X_61))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_61),Y_48)) ) ).

tff(fact_688_leD,axiom,
    ! [Y_48: int,X_61: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_48),X_61))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_61),Y_48)) ) ).

tff(fact_689_xt1_I12_J,axiom,
    ! [A_195: rat,B_150: rat] :
      ( ( A_195 != B_150 )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_150),A_195))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_150),A_195)) ) ) ).

tff(fact_690_xt1_I12_J,axiom,
    ! [Aa: fun_int_bool,Ba: fun_int_bool] :
      ( ( Aa != Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),Aa))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ba),Aa)) ) ) ).

tff(fact_691_xt1_I12_J,axiom,
    ! [Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( Aa != Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),Aa))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa)) ) ) ).

tff(fact_692_xt1_I12_J,axiom,
    ! [A_195: quickcheck_code_int,B_150: quickcheck_code_int] :
      ( ( A_195 != B_150 )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_150),A_195))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_150),A_195)) ) ) ).

tff(fact_693_xt1_I12_J,axiom,
    ! [A_195: nat,B_150: nat] :
      ( ( A_195 != B_150 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_150),A_195))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_150),A_195)) ) ) ).

tff(fact_694_xt1_I12_J,axiom,
    ! [A_195: real,B_150: real] :
      ( ( A_195 != B_150 )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_150),A_195))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_150),A_195)) ) ) ).

tff(fact_695_xt1_I12_J,axiom,
    ! [A_195: int,B_150: int] :
      ( ( A_195 != B_150 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_150),A_195))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_150),A_195)) ) ) ).

tff(fact_696_zle__refl,axiom,
    ! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).

tff(fact_697_number__of__is__id,axiom,
    ! [K_2: int] : hAPP_int_int(number_number_of_int,K_2) = K_2 ).

tff(fact_698_zle__linear,axiom,
    ! [Z_1: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) ) ).

tff(fact_699_zless__le,axiom,
    ! [Z_2: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1))
        & ( Z_2 != W_1 ) ) ) ).

tff(fact_700_zless__linear,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
      | ( X = Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).

tff(fact_701_less__number__of__int__code,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_702_less__eq__number__of__int__code,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_703_zle__trans,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K_2)) ) ) ).

tff(fact_704_zle__antisym,axiom,
    ! [Z_1: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1))
       => ( Z_1 = W ) ) ) ).

tff(fact_705_le__fun__def,axiom,
    ! [F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
    <=> ! [X_2: int] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G,X_2))) ) ).

tff(fact_706_le__fun__def,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
    <=> ! [X_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2))) ) ).

tff(fact_707_linorder__linear,axiom,
    ! [X_60: rat,Y_47: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_60),Y_47))
      | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_47),X_60)) ) ).

tff(fact_708_linorder__linear,axiom,
    ! [X_60: quickcheck_code_int,Y_47: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_60),Y_47))
      | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_47),X_60)) ) ).

tff(fact_709_linorder__linear,axiom,
    ! [X_60: nat,Y_47: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_60),Y_47))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_47),X_60)) ) ).

tff(fact_710_linorder__linear,axiom,
    ! [X_60: real,Y_47: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_60),Y_47))
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_47),X_60)) ) ).

tff(fact_711_linorder__linear,axiom,
    ! [X_60: int,Y_47: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_60),Y_47))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_47),X_60)) ) ).

tff(fact_712_order__eq__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
        & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_2),X_1)) ) ) ).

tff(fact_713_order__eq__iff,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1)) ) ) ).

tff(fact_714_order__eq__iff,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1)) ) ) ).

tff(fact_715_order__eq__iff,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
        & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_2),X_1)) ) ) ).

tff(fact_716_order__eq__iff,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) ) ) ).

tff(fact_717_order__eq__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
        & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1)) ) ) ).

tff(fact_718_order__eq__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),X_1)) ) ) ).

tff(fact_719_order__eq__refl,axiom,
    ! [X_59: rat,Y_46: rat] :
      ( ( X_59 = Y_46 )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_59),Y_46)) ) ).

tff(fact_720_order__eq__refl,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( ( X_1 = Y_2 )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2)) ) ).

tff(fact_721_order__eq__refl,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( ( X_1 = Y_2 )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2)) ) ).

tff(fact_722_order__eq__refl,axiom,
    ! [X_59: quickcheck_code_int,Y_46: quickcheck_code_int] :
      ( ( X_59 = Y_46 )
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_59),Y_46)) ) ).

tff(fact_723_order__eq__refl,axiom,
    ! [X_59: nat,Y_46: nat] :
      ( ( X_59 = Y_46 )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_59),Y_46)) ) ).

tff(fact_724_order__eq__refl,axiom,
    ! [X_59: real,Y_46: real] :
      ( ( X_59 = Y_46 )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_59),Y_46)) ) ).

tff(fact_725_order__eq__refl,axiom,
    ! [X_59: int,Y_46: int] :
      ( ( X_59 = Y_46 )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_59),Y_46)) ) ).

tff(fact_726_le__funD,axiom,
    ! [X_1: int,F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ).

tff(fact_727_le__funD,axiom,
    ! [X_1: nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).

tff(fact_728_order__antisym__conv,axiom,
    ! [Y_2: rat,X_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_2),X_1))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_729_order__antisym__conv,axiom,
    ! [Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_730_order__antisym__conv,axiom,
    ! [Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_731_order__antisym__conv,axiom,
    ! [Y_2: quickcheck_code_int,X_1: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_2),X_1))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_732_order__antisym__conv,axiom,
    ! [Y_2: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_733_order__antisym__conv,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_734_order__antisym__conv,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_735_ord__eq__le__trans,axiom,
    ! [C_71: rat,A_194: rat,B_149: rat] :
      ( ( A_194 = B_149 )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_149),C_71))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_194),C_71)) ) ) ).

tff(fact_736_ord__eq__le__trans,axiom,
    ! [C: fun_int_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),C))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Aa),C)) ) ) ).

tff(fact_737_ord__eq__le__trans,axiom,
    ! [C: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),C))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Aa),C)) ) ) ).

tff(fact_738_ord__eq__le__trans,axiom,
    ! [C_71: quickcheck_code_int,A_194: quickcheck_code_int,B_149: quickcheck_code_int] :
      ( ( A_194 = B_149 )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_149),C_71))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_194),C_71)) ) ) ).

tff(fact_739_ord__eq__le__trans,axiom,
    ! [C_71: nat,A_194: nat,B_149: nat] :
      ( ( A_194 = B_149 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_149),C_71))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_194),C_71)) ) ) ).

tff(fact_740_ord__eq__le__trans,axiom,
    ! [C_71: real,A_194: real,B_149: real] :
      ( ( A_194 = B_149 )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_149),C_71))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_194),C_71)) ) ) ).

tff(fact_741_ord__eq__le__trans,axiom,
    ! [C_71: int,A_194: int,B_149: int] :
      ( ( A_194 = B_149 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_149),C_71))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_194),C_71)) ) ) ).

tff(fact_742_xt1_I3_J,axiom,
    ! [C_70: rat,A_193: rat,B_148: rat] :
      ( ( A_193 = B_148 )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_70),B_148))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_70),A_193)) ) ) ).

tff(fact_743_xt1_I3_J,axiom,
    ! [C: fun_int_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),Ba))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),Aa)) ) ) ).

tff(fact_744_xt1_I3_J,axiom,
    ! [C: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),Ba))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),Aa)) ) ) ).

tff(fact_745_xt1_I3_J,axiom,
    ! [C_70: quickcheck_code_int,A_193: quickcheck_code_int,B_148: quickcheck_code_int] :
      ( ( A_193 = B_148 )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_70),B_148))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_70),A_193)) ) ) ).

tff(fact_746_xt1_I3_J,axiom,
    ! [C_70: nat,A_193: nat,B_148: nat] :
      ( ( A_193 = B_148 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_70),B_148))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_70),A_193)) ) ) ).

tff(fact_747_xt1_I3_J,axiom,
    ! [C_70: real,A_193: real,B_148: real] :
      ( ( A_193 = B_148 )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_70),B_148))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_70),A_193)) ) ) ).

tff(fact_748_xt1_I3_J,axiom,
    ! [C_70: int,A_193: int,B_148: int] :
      ( ( A_193 = B_148 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_70),B_148))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_70),A_193)) ) ) ).

tff(fact_749_ord__le__eq__trans,axiom,
    ! [C_69: rat,A_192: rat,B_147: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_192),B_147))
     => ( ( B_147 = C_69 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_192),C_69)) ) ) ).

tff(fact_750_ord__le__eq__trans,axiom,
    ! [C: fun_int_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Aa),Ba))
     => ( ( Ba = C )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Aa),C)) ) ) ).

tff(fact_751_ord__le__eq__trans,axiom,
    ! [C: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Aa),Ba))
     => ( ( Ba = C )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Aa),C)) ) ) ).

tff(fact_752_ord__le__eq__trans,axiom,
    ! [C_69: quickcheck_code_int,A_192: quickcheck_code_int,B_147: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_192),B_147))
     => ( ( B_147 = C_69 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_192),C_69)) ) ) ).

tff(fact_753_ord__le__eq__trans,axiom,
    ! [C_69: nat,A_192: nat,B_147: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_192),B_147))
     => ( ( B_147 = C_69 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_192),C_69)) ) ) ).

tff(fact_754_ord__le__eq__trans,axiom,
    ! [C_69: real,A_192: real,B_147: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_192),B_147))
     => ( ( B_147 = C_69 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_192),C_69)) ) ) ).

tff(fact_755_ord__le__eq__trans,axiom,
    ! [C_69: int,A_192: int,B_147: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_192),B_147))
     => ( ( B_147 = C_69 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_192),C_69)) ) ) ).

tff(fact_756_xt1_I4_J,axiom,
    ! [C_68: rat,B_146: rat,A_191: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_146),A_191))
     => ( ( B_146 = C_68 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_68),A_191)) ) ) ).

tff(fact_757_xt1_I4_J,axiom,
    ! [C: fun_int_bool,Ba: fun_int_bool,Aa: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),Aa))
     => ( ( Ba = C )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),Aa)) ) ) ).

tff(fact_758_xt1_I4_J,axiom,
    ! [C: fun_nat_bool,Ba: fun_nat_bool,Aa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),Aa))
     => ( ( Ba = C )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),Aa)) ) ) ).

tff(fact_759_xt1_I4_J,axiom,
    ! [C_68: quickcheck_code_int,B_146: quickcheck_code_int,A_191: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_146),A_191))
     => ( ( B_146 = C_68 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_68),A_191)) ) ) ).

tff(fact_760_xt1_I4_J,axiom,
    ! [C_68: nat,B_146: nat,A_191: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_146),A_191))
     => ( ( B_146 = C_68 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_68),A_191)) ) ) ).

tff(fact_761_xt1_I4_J,axiom,
    ! [C_68: real,B_146: real,A_191: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_146),A_191))
     => ( ( B_146 = C_68 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_68),A_191)) ) ) ).

tff(fact_762_xt1_I4_J,axiom,
    ! [C_68: int,B_146: int,A_191: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_146),A_191))
     => ( ( B_146 = C_68 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_68),A_191)) ) ) ).

tff(fact_763_order__antisym,axiom,
    ! [X_58: rat,Y_45: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_58),Y_45))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_45),X_58))
       => ( X_58 = Y_45 ) ) ) ).

tff(fact_764_order__antisym,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1))
       => ( X_1 = Y_2 ) ) ) ).

tff(fact_765_order__antisym,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1))
       => ( X_1 = Y_2 ) ) ) ).

tff(fact_766_order__antisym,axiom,
    ! [X_58: quickcheck_code_int,Y_45: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_58),Y_45))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_45),X_58))
       => ( X_58 = Y_45 ) ) ) ).

tff(fact_767_order__antisym,axiom,
    ! [X_58: nat,Y_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_58),Y_45))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_45),X_58))
       => ( X_58 = Y_45 ) ) ) ).

tff(fact_768_order__antisym,axiom,
    ! [X_58: real,Y_45: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_58),Y_45))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_45),X_58))
       => ( X_58 = Y_45 ) ) ) ).

tff(fact_769_order__antisym,axiom,
    ! [X_58: int,Y_45: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_58),Y_45))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_45),X_58))
       => ( X_58 = Y_45 ) ) ) ).

tff(fact_770_order__trans,axiom,
    ! [Z_15: rat,X_57: rat,Y_44: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_57),Y_44))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_44),Z_15))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_57),Z_15)) ) ) ).

tff(fact_771_order__trans,axiom,
    ! [Z_2: fun_int_bool,X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Z_2)) ) ) ).

tff(fact_772_order__trans,axiom,
    ! [Z_2: fun_nat_bool,X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Z_2)) ) ) ).

tff(fact_773_order__trans,axiom,
    ! [Z_15: quickcheck_code_int,X_57: quickcheck_code_int,Y_44: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_57),Y_44))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_44),Z_15))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_57),Z_15)) ) ) ).

tff(fact_774_order__trans,axiom,
    ! [Z_15: nat,X_57: nat,Y_44: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_57),Y_44))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_44),Z_15))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_57),Z_15)) ) ) ).

tff(fact_775_order__trans,axiom,
    ! [Z_15: real,X_57: real,Y_44: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_57),Y_44))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_44),Z_15))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_57),Z_15)) ) ) ).

tff(fact_776_order__trans,axiom,
    ! [Z_15: int,X_57: int,Y_44: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_57),Y_44))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_44),Z_15))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_57),Z_15)) ) ) ).

tff(fact_777_xt1_I5_J,axiom,
    ! [Y_43: rat,X_56: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_43),X_56))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_56),Y_43))
       => ( X_56 = Y_43 ) ) ) ).

tff(fact_778_xt1_I5_J,axiom,
    ! [Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
       => ( X_1 = Y_2 ) ) ) ).

tff(fact_779_xt1_I5_J,axiom,
    ! [Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
       => ( X_1 = Y_2 ) ) ) ).

tff(fact_780_xt1_I5_J,axiom,
    ! [Y_43: quickcheck_code_int,X_56: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_43),X_56))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_56),Y_43))
       => ( X_56 = Y_43 ) ) ) ).

tff(fact_781_xt1_I5_J,axiom,
    ! [Y_43: nat,X_56: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_43),X_56))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_56),Y_43))
       => ( X_56 = Y_43 ) ) ) ).

tff(fact_782_xt1_I5_J,axiom,
    ! [Y_43: real,X_56: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_43),X_56))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_56),Y_43))
       => ( X_56 = Y_43 ) ) ) ).

tff(fact_783_xt1_I5_J,axiom,
    ! [Y_43: int,X_56: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_43),X_56))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_56),Y_43))
       => ( X_56 = Y_43 ) ) ) ).

tff(fact_784_xt1_I6_J,axiom,
    ! [Z_14: rat,Y_42: rat,X_55: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_42),X_55))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_14),Y_42))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_14),X_55)) ) ) ).

tff(fact_785_xt1_I6_J,axiom,
    ! [Z_2: fun_int_bool,Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),X_1)) ) ) ).

tff(fact_786_xt1_I6_J,axiom,
    ! [Z_2: fun_nat_bool,Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),X_1)) ) ) ).

tff(fact_787_xt1_I6_J,axiom,
    ! [Z_14: quickcheck_code_int,Y_42: quickcheck_code_int,X_55: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_42),X_55))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_14),Y_42))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_14),X_55)) ) ) ).

tff(fact_788_xt1_I6_J,axiom,
    ! [Z_14: nat,Y_42: nat,X_55: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_42),X_55))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_14),Y_42))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_14),X_55)) ) ) ).

tff(fact_789_xt1_I6_J,axiom,
    ! [Z_14: real,Y_42: real,X_55: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_42),X_55))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_14),Y_42))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_14),X_55)) ) ) ).

tff(fact_790_xt1_I6_J,axiom,
    ! [Z_14: int,Y_42: int,X_55: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_42),X_55))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_14),Y_42))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_14),X_55)) ) ) ).

tff(fact_791_le__funE,axiom,
    ! [X_1: int,F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ).

tff(fact_792_le__funE,axiom,
    ! [X_1: nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).

tff(fact_793_linorder__le__cases,axiom,
    ! [X_54: rat,Y_41: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_54),Y_41))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_41),X_54)) ) ).

tff(fact_794_linorder__le__cases,axiom,
    ! [X_54: quickcheck_code_int,Y_41: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_54),Y_41))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_41),X_54)) ) ).

tff(fact_795_linorder__le__cases,axiom,
    ! [X_54: nat,Y_41: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_54),Y_41))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_41),X_54)) ) ).

tff(fact_796_linorder__le__cases,axiom,
    ! [X_54: real,Y_41: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_54),Y_41))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_41),X_54)) ) ).

tff(fact_797_linorder__le__cases,axiom,
    ! [X_54: int,Y_41: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_54),Y_41))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_41),X_54)) ) ).

tff(fact_798_order__less__irrefl,axiom,
    ! [X_53: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_53),X_53)) ).

tff(fact_799_order__less__irrefl,axiom,
    ! [X_1: fun_int_bool] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),X_1)) ).

tff(fact_800_order__less__irrefl,axiom,
    ! [X_1: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),X_1)) ).

tff(fact_801_order__less__irrefl,axiom,
    ! [X_53: quickcheck_code_int] : ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_53),X_53)) ).

tff(fact_802_order__less__irrefl,axiom,
    ! [X_53: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_53),X_53)) ).

tff(fact_803_order__less__irrefl,axiom,
    ! [X_53: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_53),X_53)) ).

tff(fact_804_order__less__irrefl,axiom,
    ! [X_53: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_53),X_53)) ).

tff(fact_805_linorder__neq__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ( X_1 != Y_2 )
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
        | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_2),X_1)) ) ) ).

tff(fact_806_linorder__neq__iff,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ( X_1 != Y_2 )
    <=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
        | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_2),X_1)) ) ) ).

tff(fact_807_linorder__neq__iff,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( X_1 != Y_2 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1)) ) ) ).

tff(fact_808_linorder__neq__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( X_1 != Y_2 )
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
        | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1)) ) ) ).

tff(fact_809_linorder__neq__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( X_1 != Y_2 )
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),X_1)) ) ) ).

tff(fact_810_not__less__iff__gr__or__eq,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_2),X_1))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_811_not__less__iff__gr__or__eq,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_2),X_1))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_812_not__less__iff__gr__or__eq,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_813_not__less__iff__gr__or__eq,axiom,
    ! [X_1: real,Y_2: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_814_not__less__iff__gr__or__eq,axiom,
    ! [X_1: int,Y_2: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),X_1))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_815_linorder__less__linear,axiom,
    ! [X_52: rat,Y_40: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_52),Y_40))
      | ( X_52 = Y_40 )
      | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_40),X_52)) ) ).

tff(fact_816_linorder__less__linear,axiom,
    ! [X_52: quickcheck_code_int,Y_40: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_52),Y_40))
      | ( X_52 = Y_40 )
      | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_40),X_52)) ) ).

tff(fact_817_linorder__less__linear,axiom,
    ! [X_52: nat,Y_40: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_52),Y_40))
      | ( X_52 = Y_40 )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_40),X_52)) ) ).

tff(fact_818_linorder__less__linear,axiom,
    ! [X_52: real,Y_40: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_52),Y_40))
      | ( X_52 = Y_40 )
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_40),X_52)) ) ).

tff(fact_819_linorder__less__linear,axiom,
    ! [X_52: int,Y_40: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_52),Y_40))
      | ( X_52 = Y_40 )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_40),X_52)) ) ).

tff(fact_820_linorder__antisym__conv3,axiom,
    ! [Y_2: rat,X_1: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_2),X_1))
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_821_linorder__antisym__conv3,axiom,
    ! [Y_2: quickcheck_code_int,X_1: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_2),X_1))
     => ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_822_linorder__antisym__conv3,axiom,
    ! [Y_2: nat,X_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_823_linorder__antisym__conv3,axiom,
    ! [Y_2: real,X_1: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_824_linorder__antisym__conv3,axiom,
    ! [Y_2: int,X_1: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),X_1))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_825_linorder__neqE,axiom,
    ! [X_51: rat,Y_39: rat] :
      ( ( X_51 != Y_39 )
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_51),Y_39))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_39),X_51)) ) ) ).

tff(fact_826_linorder__neqE,axiom,
    ! [X_51: quickcheck_code_int,Y_39: quickcheck_code_int] :
      ( ( X_51 != Y_39 )
     => ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_51),Y_39))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_39),X_51)) ) ) ).

tff(fact_827_linorder__neqE,axiom,
    ! [X_51: nat,Y_39: nat] :
      ( ( X_51 != Y_39 )
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_51),Y_39))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_39),X_51)) ) ) ).

tff(fact_828_linorder__neqE,axiom,
    ! [X_51: real,Y_39: real] :
      ( ( X_51 != Y_39 )
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_51),Y_39))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_39),X_51)) ) ) ).

tff(fact_829_linorder__neqE,axiom,
    ! [X_51: int,Y_39: int] :
      ( ( X_51 != Y_39 )
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_51),Y_39))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_39),X_51)) ) ) ).

tff(fact_830_less__imp__neq,axiom,
    ! [X_50: rat,Y_38: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_50),Y_38))
     => ( X_50 != Y_38 ) ) ).

tff(fact_831_less__imp__neq,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ( X_1 != Y_2 ) ) ).

tff(fact_832_less__imp__neq,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ( X_1 != Y_2 ) ) ).

tff(fact_833_less__imp__neq,axiom,
    ! [X_50: quickcheck_code_int,Y_38: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_50),Y_38))
     => ( X_50 != Y_38 ) ) ).

tff(fact_834_less__imp__neq,axiom,
    ! [X_50: nat,Y_38: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_50),Y_38))
     => ( X_50 != Y_38 ) ) ).

tff(fact_835_less__imp__neq,axiom,
    ! [X_50: real,Y_38: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_50),Y_38))
     => ( X_50 != Y_38 ) ) ).

tff(fact_836_less__imp__neq,axiom,
    ! [X_50: int,Y_38: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_50),Y_38))
     => ( X_50 != Y_38 ) ) ).

tff(fact_837_order__less__not__sym,axiom,
    ! [X_49: rat,Y_37: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_49),Y_37))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_37),X_49)) ) ).

tff(fact_838_order__less__not__sym,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1)) ) ).

tff(fact_839_order__less__not__sym,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1)) ) ).

tff(fact_840_order__less__not__sym,axiom,
    ! [X_49: quickcheck_code_int,Y_37: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_49),Y_37))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_37),X_49)) ) ).

tff(fact_841_order__less__not__sym,axiom,
    ! [X_49: nat,Y_37: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_49),Y_37))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_37),X_49)) ) ).

tff(fact_842_order__less__not__sym,axiom,
    ! [X_49: real,Y_37: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_49),Y_37))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_37),X_49)) ) ).

tff(fact_843_order__less__not__sym,axiom,
    ! [X_49: int,Y_37: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_49),Y_37))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_37),X_49)) ) ).

tff(fact_844_order__less__imp__not__less,axiom,
    ! [X_48: rat,Y_36: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_48),Y_36))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_36),X_48)) ) ).

tff(fact_845_order__less__imp__not__less,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1)) ) ).

tff(fact_846_order__less__imp__not__less,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1)) ) ).

tff(fact_847_order__less__imp__not__less,axiom,
    ! [X_48: quickcheck_code_int,Y_36: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_48),Y_36))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_36),X_48)) ) ).

tff(fact_848_order__less__imp__not__less,axiom,
    ! [X_48: nat,Y_36: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_48),Y_36))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_36),X_48)) ) ).

tff(fact_849_order__less__imp__not__less,axiom,
    ! [X_48: real,Y_36: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_48),Y_36))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_36),X_48)) ) ).

tff(fact_850_order__less__imp__not__less,axiom,
    ! [X_48: int,Y_36: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_48),Y_36))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_36),X_48)) ) ).

tff(fact_851_order__less__imp__not__eq,axiom,
    ! [X_47: rat,Y_35: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_47),Y_35))
     => ( X_47 != Y_35 ) ) ).

tff(fact_852_order__less__imp__not__eq,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ( X_1 != Y_2 ) ) ).

tff(fact_853_order__less__imp__not__eq,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ( X_1 != Y_2 ) ) ).

tff(fact_854_order__less__imp__not__eq,axiom,
    ! [X_47: quickcheck_code_int,Y_35: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_47),Y_35))
     => ( X_47 != Y_35 ) ) ).

tff(fact_855_order__less__imp__not__eq,axiom,
    ! [X_47: nat,Y_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_47),Y_35))
     => ( X_47 != Y_35 ) ) ).

tff(fact_856_order__less__imp__not__eq,axiom,
    ! [X_47: real,Y_35: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_47),Y_35))
     => ( X_47 != Y_35 ) ) ).

tff(fact_857_order__less__imp__not__eq,axiom,
    ! [X_47: int,Y_35: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_47),Y_35))
     => ( X_47 != Y_35 ) ) ).

tff(fact_858_order__less__imp__not__eq2,axiom,
    ! [X_46: rat,Y_34: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_46),Y_34))
     => ( Y_34 != X_46 ) ) ).

tff(fact_859_order__less__imp__not__eq2,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ( Y_2 != X_1 ) ) ).

tff(fact_860_order__less__imp__not__eq2,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ( Y_2 != X_1 ) ) ).

tff(fact_861_order__less__imp__not__eq2,axiom,
    ! [X_46: quickcheck_code_int,Y_34: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_46),Y_34))
     => ( Y_34 != X_46 ) ) ).

tff(fact_862_order__less__imp__not__eq2,axiom,
    ! [X_46: nat,Y_34: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_46),Y_34))
     => ( Y_34 != X_46 ) ) ).

tff(fact_863_order__less__imp__not__eq2,axiom,
    ! [X_46: real,Y_34: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_46),Y_34))
     => ( Y_34 != X_46 ) ) ).

tff(fact_864_order__less__imp__not__eq2,axiom,
    ! [X_46: int,Y_34: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_46),Y_34))
     => ( Y_34 != X_46 ) ) ).

tff(fact_865_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_866_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_867_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_868_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_869_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_870_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_871_order__less__imp__triv,axiom,
    ! [P_1: bool,X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),X_1))
       => hBOOL(P_1) ) ) ).

tff(fact_872_order__less__asym_H,axiom,
    ! [A_190: rat,B_145: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_190),B_145))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_145),A_190)) ) ).

tff(fact_873_order__less__asym_H,axiom,
    ! [Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),Ba))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ba),Aa)) ) ).

tff(fact_874_order__less__asym_H,axiom,
    ! [Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa)) ) ).

tff(fact_875_order__less__asym_H,axiom,
    ! [A_190: quickcheck_code_int,B_145: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_190),B_145))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_145),A_190)) ) ).

tff(fact_876_order__less__asym_H,axiom,
    ! [A_190: nat,B_145: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_190),B_145))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_145),A_190)) ) ).

tff(fact_877_order__less__asym_H,axiom,
    ! [A_190: real,B_145: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_190),B_145))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_145),A_190)) ) ).

tff(fact_878_order__less__asym_H,axiom,
    ! [A_190: int,B_145: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_190),B_145))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_145),A_190)) ) ).

tff(fact_879_xt1_I9_J,axiom,
    ! [B_144: rat,A_189: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_144),A_189))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_189),B_144)) ) ).

tff(fact_880_xt1_I9_J,axiom,
    ! [Ba: fun_int_bool,Aa: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ba),Aa))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),Ba)) ) ).

tff(fact_881_xt1_I9_J,axiom,
    ! [Ba: fun_nat_bool,Aa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba)) ) ).

tff(fact_882_xt1_I9_J,axiom,
    ! [B_144: quickcheck_code_int,A_189: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_144),A_189))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_189),B_144)) ) ).

tff(fact_883_xt1_I9_J,axiom,
    ! [B_144: nat,A_189: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_144),A_189))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_189),B_144)) ) ).

tff(fact_884_xt1_I9_J,axiom,
    ! [B_144: real,A_189: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_144),A_189))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_189),B_144)) ) ).

tff(fact_885_xt1_I9_J,axiom,
    ! [B_144: int,A_189: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_144),A_189))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_189),B_144)) ) ).

tff(fact_886_ord__eq__less__trans,axiom,
    ! [C_67: rat,A_188: rat,B_143: rat] :
      ( ( A_188 = B_143 )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_143),C_67))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_188),C_67)) ) ) ).

tff(fact_887_ord__eq__less__trans,axiom,
    ! [C: fun_int_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ba),C))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),C)) ) ) ).

tff(fact_888_ord__eq__less__trans,axiom,
    ! [C: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),C))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),C)) ) ) ).

tff(fact_889_ord__eq__less__trans,axiom,
    ! [C_67: quickcheck_code_int,A_188: quickcheck_code_int,B_143: quickcheck_code_int] :
      ( ( A_188 = B_143 )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_143),C_67))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_188),C_67)) ) ) ).

tff(fact_890_ord__eq__less__trans,axiom,
    ! [C_67: nat,A_188: nat,B_143: nat] :
      ( ( A_188 = B_143 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_143),C_67))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_188),C_67)) ) ) ).

tff(fact_891_ord__eq__less__trans,axiom,
    ! [C_67: real,A_188: real,B_143: real] :
      ( ( A_188 = B_143 )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_143),C_67))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_188),C_67)) ) ) ).

tff(fact_892_ord__eq__less__trans,axiom,
    ! [C_67: int,A_188: int,B_143: int] :
      ( ( A_188 = B_143 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_143),C_67))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_188),C_67)) ) ) ).

tff(fact_893_xt1_I1_J,axiom,
    ! [C_66: rat,A_187: rat,B_142: rat] :
      ( ( A_187 = B_142 )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_66),B_142))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_66),A_187)) ) ) ).

tff(fact_894_xt1_I1_J,axiom,
    ! [C: fun_int_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),Ba))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),Aa)) ) ) ).

tff(fact_895_xt1_I1_J,axiom,
    ! [C: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( Aa = Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),Ba))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),Aa)) ) ) ).

tff(fact_896_xt1_I1_J,axiom,
    ! [C_66: quickcheck_code_int,A_187: quickcheck_code_int,B_142: quickcheck_code_int] :
      ( ( A_187 = B_142 )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_66),B_142))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_66),A_187)) ) ) ).

tff(fact_897_xt1_I1_J,axiom,
    ! [C_66: nat,A_187: nat,B_142: nat] :
      ( ( A_187 = B_142 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_66),B_142))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_66),A_187)) ) ) ).

tff(fact_898_xt1_I1_J,axiom,
    ! [C_66: real,A_187: real,B_142: real] :
      ( ( A_187 = B_142 )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_66),B_142))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_66),A_187)) ) ) ).

tff(fact_899_xt1_I1_J,axiom,
    ! [C_66: int,A_187: int,B_142: int] :
      ( ( A_187 = B_142 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_66),B_142))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_66),A_187)) ) ) ).

tff(fact_900_ord__less__eq__trans,axiom,
    ! [C_65: rat,A_186: rat,B_141: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_186),B_141))
     => ( ( B_141 = C_65 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_186),C_65)) ) ) ).

tff(fact_901_ord__less__eq__trans,axiom,
    ! [C: fun_int_bool,Aa: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),Ba))
     => ( ( Ba = C )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),C)) ) ) ).

tff(fact_902_ord__less__eq__trans,axiom,
    ! [C: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba))
     => ( ( Ba = C )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),C)) ) ) ).

tff(fact_903_ord__less__eq__trans,axiom,
    ! [C_65: quickcheck_code_int,A_186: quickcheck_code_int,B_141: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_186),B_141))
     => ( ( B_141 = C_65 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_186),C_65)) ) ) ).

tff(fact_904_ord__less__eq__trans,axiom,
    ! [C_65: nat,A_186: nat,B_141: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_186),B_141))
     => ( ( B_141 = C_65 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_186),C_65)) ) ) ).

tff(fact_905_ord__less__eq__trans,axiom,
    ! [C_65: real,A_186: real,B_141: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_186),B_141))
     => ( ( B_141 = C_65 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_186),C_65)) ) ) ).

tff(fact_906_ord__less__eq__trans,axiom,
    ! [C_65: int,A_186: int,B_141: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_186),B_141))
     => ( ( B_141 = C_65 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_186),C_65)) ) ) ).

tff(fact_907_xt1_I2_J,axiom,
    ! [C_64: rat,B_140: rat,A_185: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_140),A_185))
     => ( ( B_140 = C_64 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_64),A_185)) ) ) ).

tff(fact_908_xt1_I2_J,axiom,
    ! [C: fun_int_bool,Ba: fun_int_bool,Aa: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ba),Aa))
     => ( ( Ba = C )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),Aa)) ) ) ).

tff(fact_909_xt1_I2_J,axiom,
    ! [C: fun_nat_bool,Ba: fun_nat_bool,Aa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa))
     => ( ( Ba = C )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),Aa)) ) ) ).

tff(fact_910_xt1_I2_J,axiom,
    ! [C_64: quickcheck_code_int,B_140: quickcheck_code_int,A_185: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_140),A_185))
     => ( ( B_140 = C_64 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_64),A_185)) ) ) ).

tff(fact_911_xt1_I2_J,axiom,
    ! [C_64: nat,B_140: nat,A_185: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_140),A_185))
     => ( ( B_140 = C_64 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_64),A_185)) ) ) ).

tff(fact_912_xt1_I2_J,axiom,
    ! [C_64: real,B_140: real,A_185: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_140),A_185))
     => ( ( B_140 = C_64 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_64),A_185)) ) ) ).

tff(fact_913_xt1_I2_J,axiom,
    ! [C_64: int,B_140: int,A_185: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_140),A_185))
     => ( ( B_140 = C_64 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_64),A_185)) ) ) ).

tff(fact_914_order__less__trans,axiom,
    ! [Z_13: rat,X_45: rat,Y_33: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_45),Y_33))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_33),Z_13))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_45),Z_13)) ) ) ).

tff(fact_915_order__less__trans,axiom,
    ! [Z_2: fun_int_bool,X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Z_2)) ) ) ).

tff(fact_916_order__less__trans,axiom,
    ! [Z_2: fun_nat_bool,X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),Z_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Z_2)) ) ) ).

tff(fact_917_order__less__trans,axiom,
    ! [Z_13: quickcheck_code_int,X_45: quickcheck_code_int,Y_33: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_45),Y_33))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_33),Z_13))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_45),Z_13)) ) ) ).

tff(fact_918_order__less__trans,axiom,
    ! [Z_13: nat,X_45: nat,Y_33: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_45),Y_33))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_33),Z_13))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_45),Z_13)) ) ) ).

tff(fact_919_order__less__trans,axiom,
    ! [Z_13: real,X_45: real,Y_33: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_45),Y_33))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_33),Z_13))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_45),Z_13)) ) ) ).

tff(fact_920_order__less__trans,axiom,
    ! [Z_13: int,X_45: int,Y_33: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_45),Y_33))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_33),Z_13))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_45),Z_13)) ) ) ).

tff(fact_921_xt1_I10_J,axiom,
    ! [Z_12: rat,Y_32: rat,X_44: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_32),X_44))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_12),Y_32))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_12),X_44)) ) ) ).

tff(fact_922_xt1_I10_J,axiom,
    ! [Z_2: fun_int_bool,Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),X_1)) ) ) ).

tff(fact_923_xt1_I10_J,axiom,
    ! [Z_2: fun_nat_bool,Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Y_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),X_1)) ) ) ).

tff(fact_924_xt1_I10_J,axiom,
    ! [Z_12: quickcheck_code_int,Y_32: quickcheck_code_int,X_44: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_32),X_44))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_12),Y_32))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_12),X_44)) ) ) ).

tff(fact_925_xt1_I10_J,axiom,
    ! [Z_12: nat,Y_32: nat,X_44: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_32),X_44))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_12),Y_32))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_12),X_44)) ) ) ).

tff(fact_926_xt1_I10_J,axiom,
    ! [Z_12: real,Y_32: real,X_44: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_32),X_44))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_12),Y_32))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_12),X_44)) ) ) ).

tff(fact_927_xt1_I10_J,axiom,
    ! [Z_12: int,Y_32: int,X_44: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_32),X_44))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_12),Y_32))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_12),X_44)) ) ) ).

tff(fact_928_order__less__asym,axiom,
    ! [X_43: rat,Y_31: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_43),Y_31))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_31),X_43)) ) ).

tff(fact_929_order__less__asym,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1)) ) ).

tff(fact_930_order__less__asym,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1)) ) ).

tff(fact_931_order__less__asym,axiom,
    ! [X_43: quickcheck_code_int,Y_31: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_43),Y_31))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_31),X_43)) ) ).

tff(fact_932_order__less__asym,axiom,
    ! [X_43: nat,Y_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_43),Y_31))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_31),X_43)) ) ).

tff(fact_933_order__less__asym,axiom,
    ! [X_43: real,Y_31: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_43),Y_31))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_31),X_43)) ) ).

tff(fact_934_order__less__asym,axiom,
    ! [X_43: int,Y_31: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_43),Y_31))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_31),X_43)) ) ).

tff(fact_935_linorder__cases,axiom,
    ! [X_42: rat,Y_30: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_42),Y_30))
     => ( ( X_42 != Y_30 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_30),X_42)) ) ) ).

tff(fact_936_linorder__cases,axiom,
    ! [X_42: quickcheck_code_int,Y_30: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_42),Y_30))
     => ( ( X_42 != Y_30 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_30),X_42)) ) ) ).

tff(fact_937_linorder__cases,axiom,
    ! [X_42: nat,Y_30: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_42),Y_30))
     => ( ( X_42 != Y_30 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_30),X_42)) ) ) ).

tff(fact_938_linorder__cases,axiom,
    ! [X_42: real,Y_30: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_42),Y_30))
     => ( ( X_42 != Y_30 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_30),X_42)) ) ) ).

tff(fact_939_linorder__cases,axiom,
    ! [X_42: int,Y_30: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_42),Y_30))
     => ( ( X_42 != Y_30 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_30),X_42)) ) ) ).

tff(fact_940_linorder__not__less,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_2),X_1)) ) ).

tff(fact_941_linorder__not__less,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_2),X_1)) ) ).

tff(fact_942_linorder__not__less,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) ) ).

tff(fact_943_linorder__not__less,axiom,
    ! [X_1: real,Y_2: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1)) ) ).

tff(fact_944_linorder__not__less,axiom,
    ! [X_1: int,Y_2: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),X_1)) ) ).

tff(fact_945_linorder__not__le,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_2),X_1)) ) ).

tff(fact_946_linorder__not__le,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_2),X_1)) ) ).

tff(fact_947_linorder__not__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1)) ) ).

tff(fact_948_linorder__not__le,axiom,
    ! [X_1: real,Y_2: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1)) ) ).

tff(fact_949_linorder__not__le,axiom,
    ! [X_1: int,Y_2: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),X_1)) ) ).

tff(fact_950_linorder__le__less__linear,axiom,
    ! [X_41: rat,Y_29: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_41),Y_29))
      | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_29),X_41)) ) ).

tff(fact_951_linorder__le__less__linear,axiom,
    ! [X_41: quickcheck_code_int,Y_29: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_41),Y_29))
      | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_29),X_41)) ) ).

tff(fact_952_linorder__le__less__linear,axiom,
    ! [X_41: nat,Y_29: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_41),Y_29))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_29),X_41)) ) ).

tff(fact_953_linorder__le__less__linear,axiom,
    ! [X_41: real,Y_29: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_41),Y_29))
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_29),X_41)) ) ).

tff(fact_954_linorder__le__less__linear,axiom,
    ! [X_41: int,Y_29: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_41),Y_29))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_29),X_41)) ) ).

tff(fact_955_order__less__le,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_956_order__less__le,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_957_order__less__le,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_958_order__less__le,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_959_order__less__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_960_order__less__le,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_961_order__less__le,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_962_less__le__not__le,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
        & ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_2),X_1)) ) ) ).

tff(fact_963_less__le__not__le,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
        & ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1)) ) ) ).

tff(fact_964_less__le__not__le,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1)) ) ) ).

tff(fact_965_less__le__not__le,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
        & ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_2),X_1)) ) ) ).

tff(fact_966_less__le__not__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) ) ) ).

tff(fact_967_less__le__not__le,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
        & ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1)) ) ) ).

tff(fact_968_less__le__not__le,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
        & ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),X_1)) ) ) ).

tff(fact_969_order__le__less,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_970_order__le__less,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_971_order__le__less,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_972_order__le__less,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_973_order__le__less,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_974_order__le__less,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_975_order__le__less,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_976_leI,axiom,
    ! [X_40: rat,Y_28: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_40),Y_28))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_28),X_40)) ) ).

tff(fact_977_leI,axiom,
    ! [X_40: quickcheck_code_int,Y_28: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_40),Y_28))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_28),X_40)) ) ).

tff(fact_978_leI,axiom,
    ! [X_40: nat,Y_28: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_40),Y_28))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_28),X_40)) ) ).

tff(fact_979_leI,axiom,
    ! [X_40: real,Y_28: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_40),Y_28))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_28),X_40)) ) ).

tff(fact_980_leI,axiom,
    ! [X_40: int,Y_28: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_40),Y_28))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_28),X_40)) ) ).

tff(fact_981_not__leE,axiom,
    ! [Y_27: rat,X_39: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_27),X_39))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_39),Y_27)) ) ).

tff(fact_982_not__leE,axiom,
    ! [Y_27: quickcheck_code_int,X_39: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_27),X_39))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_39),Y_27)) ) ).

tff(fact_983_not__leE,axiom,
    ! [Y_27: nat,X_39: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_27),X_39))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_39),Y_27)) ) ).

tff(fact_984_not__leE,axiom,
    ! [Y_27: real,X_39: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_27),X_39))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_39),Y_27)) ) ).

tff(fact_985_not__leE,axiom,
    ! [Y_27: int,X_39: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_27),X_39))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_39),Y_27)) ) ).

tff(fact_986_linorder__antisym__conv1,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_987_linorder__antisym__conv1,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_1),Y_2))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_988_linorder__antisym__conv1,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_989_linorder__antisym__conv1,axiom,
    ! [X_1: real,Y_2: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_990_linorder__antisym__conv1,axiom,
    ! [X_1: int,Y_2: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_991_order__neq__le__trans,axiom,
    ! [A_184: rat,B_139: rat] :
      ( ( A_184 != B_139 )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_184),B_139))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_184),B_139)) ) ) ).

tff(fact_992_order__neq__le__trans,axiom,
    ! [Aa: fun_int_bool,Ba: fun_int_bool] :
      ( ( Aa != Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Aa),Ba))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Aa),Ba)) ) ) ).

tff(fact_993_order__neq__le__trans,axiom,
    ! [Aa: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( Aa != Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Aa),Ba))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba)) ) ) ).

tff(fact_994_order__neq__le__trans,axiom,
    ! [A_184: quickcheck_code_int,B_139: quickcheck_code_int] :
      ( ( A_184 != B_139 )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_184),B_139))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_184),B_139)) ) ) ).

tff(fact_995_order__neq__le__trans,axiom,
    ! [A_184: nat,B_139: nat] :
      ( ( A_184 != B_139 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_184),B_139))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_184),B_139)) ) ) ).

tff(fact_996_order__neq__le__trans,axiom,
    ! [A_184: real,B_139: real] :
      ( ( A_184 != B_139 )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_184),B_139))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_184),B_139)) ) ) ).

tff(fact_997_order__neq__le__trans,axiom,
    ! [A_184: int,B_139: int] :
      ( ( A_184 != B_139 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_184),B_139))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_184),B_139)) ) ) ).

tff(fact_998_norR__mem__unique__aux,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),one_one_int)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1)) ) ).

tff(fact_999_int__le__induct,axiom,
    ! [P_1: fun_int_bool,I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
     => ( hBOOL(hAPP_int_bool(P_1,K_1))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_1000_int__less__induct,axiom,
    ! [P_1: fun_int_bool,I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
     => ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int)))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_1001_int__ge__induct,axiom,
    ! [P_1: fun_int_bool,K_1: int,I_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
     => ( hBOOL(hAPP_int_bool(P_1,K_1))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_1002_int__gr__induct,axiom,
    ! [P_1: fun_int_bool,K_1: int,I_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
     => ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int)))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_1003_dense__le,axiom,
    ! [Z_11: rat,Y_26: rat] :
      ( ! [X_2: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Y_26))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_2),Z_11)) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_26),Z_11)) ) ).

tff(fact_1004_dense__le,axiom,
    ! [Z_11: real,Y_26: real] :
      ( ! [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_26))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_11)) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_26),Z_11)) ) ).

tff(fact_1005_dense__le__bounded,axiom,
    ! [Z_10: rat,X_38: rat,Y_25: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_38),Y_25))
     => ( ! [W_8: rat] :
            ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_38),W_8))
           => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,W_8),Y_25))
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,W_8),Z_10)) ) )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_25),Z_10)) ) ) ).

tff(fact_1006_dense__le__bounded,axiom,
    ! [Z_10: real,X_38: real,Y_25: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_38),Y_25))
     => ( ! [W_8: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_38),W_8))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,W_8),Y_25))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_8),Z_10)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_25),Z_10)) ) ) ).

tff(fact_1007_less__fun__def,axiom,
    ! [F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,F),G))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
        & ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,G),F)) ) ) ).

tff(fact_1008_less__fun__def,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,F),G))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,G),F)) ) ) ).

tff(fact_1009_le__funI,axiom,
    ! [F: fun_int_bool,G: fun_int_bool] :
      ( ! [X_2: int] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G,X_2)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) ) ).

tff(fact_1010_le__funI,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool] :
      ( ! [X_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)) ) ).

tff(fact_1011_int__induct,axiom,
    ! [I_1: int,P_1: fun_int_bool,K_1: int] :
      ( hBOOL(hAPP_int_bool(P_1,K_1))
     => ( ! [I_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
           => ( hBOOL(hAPP_int_bool(P_1,I_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_1012_Euler_Oaux2,axiom,
    ! [B_1: int,A_2: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),C_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_1))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_2)) ) ) ) ).

tff(fact_1013_d22set__induct__old,axiom,
    ! [X_1: int,P_1: fun_int_bool] :
      ( ! [A_1: int] :
          ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),one_one_int))) )
         => hBOOL(hAPP_int_bool(P_1,A_1)) )
     => hBOOL(hAPP_int_bool(P_1,X_1)) ) ).

tff(fact_1014_convex__bound__lt,axiom,
    ! [V_13: rat,U_3: rat,Y_24: rat,X_37: rat,A_183: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_37),A_183))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_24),A_183))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_3))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_13))
           => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),V_13) = one_one_rat )
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_3),X_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_13),Y_24))),A_183)) ) ) ) ) ) ).

tff(fact_1015_convex__bound__lt,axiom,
    ! [V_13: real,U_3: real,Y_24: real,X_37: real,A_183: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_37),A_183))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_24),A_183))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_13))
           => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),V_13) = one_one_real )
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_3),X_37)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_13),Y_24))),A_183)) ) ) ) ) ) ).

tff(fact_1016_convex__bound__lt,axiom,
    ! [V_13: int,U_3: int,Y_24: int,X_37: int,A_183: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),A_183))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),A_183))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_13))
           => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_13) = one_one_int )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_3),X_37)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_13),Y_24))),A_183)) ) ) ) ) ) ).

tff(fact_1017_number__of__le__ceiling,axiom,
    ! [V: int,X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),archim1419379755ng_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V)),one_one_rat)),X_1)) ) ).

tff(fact_1018_number__of__le__ceiling,axiom,
    ! [V: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V)),one_one_real)),X_1)) ) ).

tff(fact_1019_ceiling__less__number__of,axiom,
    ! [X_1: rat,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1419379755ng_rat(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V)),one_one_rat))) ) ).

tff(fact_1020_ceiling__less__number__of,axiom,
    ! [X_1: real,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V)),one_one_real))) ) ).

tff(fact_1021_zero__less__ceiling,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),archim1419379755ng_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),X_1)) ) ).

tff(fact_1022_zero__less__ceiling,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_1023_ceiling__le__zero,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1419379755ng_rat(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),zero_zero_rat)) ) ).

tff(fact_1024_ceiling__le__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).

tff(fact_1025_ceiling__zero,axiom,
    archim1419379755ng_rat(zero_zero_rat) = zero_zero_int ).

tff(fact_1026_ceiling__zero,axiom,
    archim856651990g_real(zero_zero_real) = zero_zero_int ).

tff(fact_1027_zero__reorient,axiom,
    ! [X_1: rat] :
      ( ( zero_zero_rat = X_1 )
    <=> ( X_1 = zero_zero_rat ) ) ).

tff(fact_1028_zero__reorient,axiom,
    ! [X_1: quickcheck_code_int] :
      ( ( zero_z891286103de_int = X_1 )
    <=> ( X_1 = zero_z891286103de_int ) ) ).

tff(fact_1029_zero__reorient,axiom,
    ! [X_1: complex] :
      ( ( zero_zero_complex = X_1 )
    <=> ( X_1 = zero_zero_complex ) ) ).

tff(fact_1030_zero__reorient,axiom,
    ! [X_1: code_code_numeral] :
      ( ( zero_z126310315umeral = X_1 )
    <=> ( X_1 = zero_z126310315umeral ) ) ).

tff(fact_1031_zero__reorient,axiom,
    ! [X_1: nat] :
      ( ( zero_zero_nat = X_1 )
    <=> ( X_1 = zero_zero_nat ) ) ).

tff(fact_1032_zero__reorient,axiom,
    ! [X_1: real] :
      ( ( zero_zero_real = X_1 )
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_1033_zero__reorient,axiom,
    ! [X_1: int] :
      ( ( zero_zero_int = X_1 )
    <=> ( X_1 = zero_zero_int ) ) ).

tff(fact_1034_ceiling__less__one,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1419379755ng_rat(X_1)),one_one_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),zero_zero_rat)) ) ).

tff(fact_1035_ceiling__less__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),one_one_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).

tff(fact_1036_one__le__ceiling,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),archim1419379755ng_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),X_1)) ) ).

tff(fact_1037_one__le__ceiling,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_1038_divisors__zero,axiom,
    ! [A_182: rat,B_138: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_182),B_138) = zero_zero_rat )
     => ( ( A_182 = zero_zero_rat )
        | ( B_138 = zero_zero_rat ) ) ) ).

tff(fact_1039_divisors__zero,axiom,
    ! [A_182: quickcheck_code_int,B_138: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_182),B_138) = zero_z891286103de_int )
     => ( ( A_182 = zero_z891286103de_int )
        | ( B_138 = zero_z891286103de_int ) ) ) ).

tff(fact_1040_divisors__zero,axiom,
    ! [A_182: complex,B_138: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_182),B_138) = zero_zero_complex )
     => ( ( A_182 = zero_zero_complex )
        | ( B_138 = zero_zero_complex ) ) ) ).

tff(fact_1041_divisors__zero,axiom,
    ! [A_182: code_code_numeral,B_138: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_182),B_138) = zero_z126310315umeral )
     => ( ( A_182 = zero_z126310315umeral )
        | ( B_138 = zero_z126310315umeral ) ) ) ).

tff(fact_1042_divisors__zero,axiom,
    ! [A_182: nat,B_138: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_182),B_138) = zero_zero_nat )
     => ( ( A_182 = zero_zero_nat )
        | ( B_138 = zero_zero_nat ) ) ) ).

tff(fact_1043_divisors__zero,axiom,
    ! [A_182: real,B_138: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_182),B_138) = zero_zero_real )
     => ( ( A_182 = zero_zero_real )
        | ( B_138 = zero_zero_real ) ) ) ).

tff(fact_1044_divisors__zero,axiom,
    ! [A_182: int,B_138: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_182),B_138) = zero_zero_int )
     => ( ( A_182 = zero_zero_int )
        | ( B_138 = zero_zero_int ) ) ) ).

tff(fact_1045_no__zero__divisors,axiom,
    ! [B_137: rat,A_181: rat] :
      ( ( A_181 != zero_zero_rat )
     => ( ( B_137 != zero_zero_rat )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_181),B_137) != zero_zero_rat ) ) ) ).

tff(fact_1046_no__zero__divisors,axiom,
    ! [B_137: quickcheck_code_int,A_181: quickcheck_code_int] :
      ( ( A_181 != zero_z891286103de_int )
     => ( ( B_137 != zero_z891286103de_int )
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_181),B_137) != zero_z891286103de_int ) ) ) ).

tff(fact_1047_no__zero__divisors,axiom,
    ! [B_137: complex,A_181: complex] :
      ( ( A_181 != zero_zero_complex )
     => ( ( B_137 != zero_zero_complex )
       => ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_181),B_137) != zero_zero_complex ) ) ) ).

tff(fact_1048_no__zero__divisors,axiom,
    ! [B_137: code_code_numeral,A_181: code_code_numeral] :
      ( ( A_181 != zero_z126310315umeral )
     => ( ( B_137 != zero_z126310315umeral )
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_181),B_137) != zero_z126310315umeral ) ) ) ).

tff(fact_1049_no__zero__divisors,axiom,
    ! [B_137: nat,A_181: nat] :
      ( ( A_181 != zero_zero_nat )
     => ( ( B_137 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_181),B_137) != zero_zero_nat ) ) ) ).

tff(fact_1050_no__zero__divisors,axiom,
    ! [B_137: real,A_181: real] :
      ( ( A_181 != zero_zero_real )
     => ( ( B_137 != zero_zero_real )
       => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_181),B_137) != zero_zero_real ) ) ) ).

tff(fact_1051_no__zero__divisors,axiom,
    ! [B_137: int,A_181: int] :
      ( ( A_181 != zero_zero_int )
     => ( ( B_137 != zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_181),B_137) != zero_zero_int ) ) ) ).

tff(fact_1052_mult__eq__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),Ba) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        | ( Ba = zero_zero_rat ) ) ) ).

tff(fact_1053_mult__eq__0__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),Ba) = zero_zero_complex )
    <=> ( ( Aa = zero_zero_complex )
        | ( Ba = zero_zero_complex ) ) ) ).

tff(fact_1054_mult__eq__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),Ba) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        | ( Ba = zero_zero_real ) ) ) ).

tff(fact_1055_mult__eq__0__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba) = zero_zero_int )
    <=> ( ( Aa = zero_zero_int )
        | ( Ba = zero_zero_int ) ) ) ).

tff(fact_1056_mult__right_Ozero,axiom,
    ! [X_36: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),zero_zero_complex) = zero_zero_complex ).

tff(fact_1057_mult__right_Ozero,axiom,
    ! [X_36: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),zero_zero_real) = zero_zero_real ).

tff(fact_1058_mult_Ozero__right,axiom,
    ! [A_180: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_180),zero_zero_complex) = zero_zero_complex ).

tff(fact_1059_mult_Ozero__right,axiom,
    ! [A_180: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_180),zero_zero_real) = zero_zero_real ).

tff(fact_1060_mult__zero__right,axiom,
    ! [A_179: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_179),zero_zero_rat) = zero_zero_rat ).

tff(fact_1061_mult__zero__right,axiom,
    ! [A_179: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_179),zero_z891286103de_int) = zero_z891286103de_int ).

tff(fact_1062_mult__zero__right,axiom,
    ! [A_179: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_179),zero_zero_complex) = zero_zero_complex ).

tff(fact_1063_mult__zero__right,axiom,
    ! [A_179: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_179),zero_z126310315umeral) = zero_z126310315umeral ).

tff(fact_1064_mult__zero__right,axiom,
    ! [A_179: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_179),zero_zero_nat) = zero_zero_nat ).

tff(fact_1065_mult__zero__right,axiom,
    ! [A_179: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_179),zero_zero_real) = zero_zero_real ).

tff(fact_1066_mult__zero__right,axiom,
    ! [A_179: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_179),zero_zero_int) = zero_zero_int ).

tff(fact_1067_mult__left_Ozero,axiom,
    ! [Y_23: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),Y_23) = zero_zero_complex ).

tff(fact_1068_mult__left_Ozero,axiom,
    ! [Y_23: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),Y_23) = zero_zero_real ).

tff(fact_1069_mult_Ozero__left,axiom,
    ! [B_136: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),B_136) = zero_zero_complex ).

tff(fact_1070_mult_Ozero__left,axiom,
    ! [B_136: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),B_136) = zero_zero_real ).

tff(fact_1071_mult__zero__left,axiom,
    ! [A_178: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_178) = zero_zero_rat ).

tff(fact_1072_mult__zero__left,axiom,
    ! [A_178: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_178) = zero_z891286103de_int ).

tff(fact_1073_mult__zero__left,axiom,
    ! [A_178: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_178) = zero_zero_complex ).

tff(fact_1074_mult__zero__left,axiom,
    ! [A_178: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_178) = zero_z126310315umeral ).

tff(fact_1075_mult__zero__left,axiom,
    ! [A_178: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_178) = zero_zero_nat ).

tff(fact_1076_mult__zero__left,axiom,
    ! [A_178: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_178) = zero_zero_real ).

tff(fact_1077_mult__zero__left,axiom,
    ! [A_178: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_178) = zero_zero_int ).

tff(fact_1078_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_177),zero_zero_rat) = zero_zero_rat ).

tff(fact_1079_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_177),zero_z891286103de_int) = zero_z891286103de_int ).

tff(fact_1080_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_177),zero_zero_complex) = zero_zero_complex ).

tff(fact_1081_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_177),zero_z126310315umeral) = zero_z126310315umeral ).

tff(fact_1082_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_177),zero_zero_nat) = zero_zero_nat ).

tff(fact_1083_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_177),zero_zero_real) = zero_zero_real ).

tff(fact_1084_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_177: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_177),zero_zero_int) = zero_zero_int ).

tff(fact_1085_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_176) = zero_zero_rat ).

tff(fact_1086_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_176) = zero_z891286103de_int ).

tff(fact_1087_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_176) = zero_zero_complex ).

tff(fact_1088_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_176) = zero_z126310315umeral ).

tff(fact_1089_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_176) = zero_zero_nat ).

tff(fact_1090_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_176) = zero_zero_real ).

tff(fact_1091_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_176: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_176) = zero_zero_int ).

tff(fact_1092_double__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1093_double__eq__0__iff,axiom,
    ! [Aa: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1094_double__eq__0__iff,axiom,
    ! [Aa: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1095_add_Ocomm__neutral,axiom,
    ! [A_175: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_175),zero_zero_rat) = A_175 ).

tff(fact_1096_add_Ocomm__neutral,axiom,
    ! [A_175: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_175),zero_z891286103de_int) = A_175 ).

tff(fact_1097_add_Ocomm__neutral,axiom,
    ! [A_175: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_175),zero_zero_complex) = A_175 ).

tff(fact_1098_add_Ocomm__neutral,axiom,
    ! [A_175: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_175),zero_zero_nat) = A_175 ).

tff(fact_1099_add_Ocomm__neutral,axiom,
    ! [A_175: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_175),zero_zero_real) = A_175 ).

tff(fact_1100_add_Ocomm__neutral,axiom,
    ! [A_175: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_175),zero_zero_int) = A_175 ).

tff(fact_1101_add_Ocomm__neutral,axiom,
    ! [A_175: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_175),zero_z126310315umeral) = A_175 ).

tff(fact_1102_add__0__right,axiom,
    ! [A_174: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_174),zero_zero_rat) = A_174 ).

tff(fact_1103_add__0__right,axiom,
    ! [A_174: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_174),zero_z891286103de_int) = A_174 ).

tff(fact_1104_add__0__right,axiom,
    ! [A_174: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_174),zero_zero_complex) = A_174 ).

tff(fact_1105_add__0__right,axiom,
    ! [A_174: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_174),zero_zero_nat) = A_174 ).

tff(fact_1106_add__0__right,axiom,
    ! [A_174: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_174),zero_zero_real) = A_174 ).

tff(fact_1107_add__0__right,axiom,
    ! [A_174: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_174),zero_zero_int) = A_174 ).

tff(fact_1108_add__0__right,axiom,
    ! [A_174: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_174),zero_z126310315umeral) = A_174 ).

tff(fact_1109_double__zero__sym,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1110_double__zero__sym,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1111_double__zero__sym,axiom,
    ! [Aa: int] :
      ( ( zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1112_add__0,axiom,
    ! [A_173: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_173) = A_173 ).

tff(fact_1113_add__0,axiom,
    ! [A_173: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_173) = A_173 ).

tff(fact_1114_add__0,axiom,
    ! [A_173: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_173) = A_173 ).

tff(fact_1115_add__0,axiom,
    ! [A_173: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_173) = A_173 ).

tff(fact_1116_add__0,axiom,
    ! [A_173: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_173) = A_173 ).

tff(fact_1117_add__0,axiom,
    ! [A_173: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_173) = A_173 ).

tff(fact_1118_add__0,axiom,
    ! [A_173: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_173) = A_173 ).

tff(fact_1119_add__0__left,axiom,
    ! [A_172: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_172) = A_172 ).

tff(fact_1120_add__0__left,axiom,
    ! [A_172: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_172) = A_172 ).

tff(fact_1121_add__0__left,axiom,
    ! [A_172: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_172) = A_172 ).

tff(fact_1122_add__0__left,axiom,
    ! [A_172: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_172) = A_172 ).

tff(fact_1123_add__0__left,axiom,
    ! [A_172: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_172) = A_172 ).

tff(fact_1124_add__0__left,axiom,
    ! [A_172: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_172) = A_172 ).

tff(fact_1125_add__0__left,axiom,
    ! [A_172: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_172) = A_172 ).

tff(fact_1126_add__0__iff,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Ba),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1127_add__0__iff,axiom,
    ! [Ba: complex,Aa: complex] :
      ( ( Ba = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Ba),Aa) )
    <=> ( Aa = zero_zero_complex ) ) ).

tff(fact_1128_add__0__iff,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) )
    <=> ( Aa = zero_zero_nat ) ) ).

tff(fact_1129_add__0__iff,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Ba),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1130_add__0__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1131_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_171),zero_zero_rat) = A_171 ).

tff(fact_1132_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_171),zero_z891286103de_int) = A_171 ).

tff(fact_1133_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_171),zero_zero_complex) = A_171 ).

tff(fact_1134_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_171),zero_zero_nat) = A_171 ).

tff(fact_1135_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_171),zero_zero_real) = A_171 ).

tff(fact_1136_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_171),zero_zero_int) = A_171 ).

tff(fact_1137_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_171: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_171),zero_z126310315umeral) = A_171 ).

tff(fact_1138_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_170) = A_170 ).

tff(fact_1139_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_170) = A_170 ).

tff(fact_1140_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_170) = A_170 ).

tff(fact_1141_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_170) = A_170 ).

tff(fact_1142_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_170) = A_170 ).

tff(fact_1143_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_170) = A_170 ).

tff(fact_1144_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_170: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_170) = A_170 ).

tff(fact_1145_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

tff(fact_1146_zero__neq__one,axiom,
    zero_z891286103de_int != one_on1684967323de_int ).

tff(fact_1147_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

tff(fact_1148_zero__neq__one,axiom,
    zero_z126310315umeral != one_on1645066479umeral ).

tff(fact_1149_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

tff(fact_1150_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

tff(fact_1151_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

tff(fact_1152_one__neq__zero,axiom,
    one_one_rat != zero_zero_rat ).

tff(fact_1153_one__neq__zero,axiom,
    one_on1684967323de_int != zero_z891286103de_int ).

tff(fact_1154_one__neq__zero,axiom,
    one_one_complex != zero_zero_complex ).

tff(fact_1155_one__neq__zero,axiom,
    one_on1645066479umeral != zero_z126310315umeral ).

tff(fact_1156_one__neq__zero,axiom,
    one_one_nat != zero_zero_nat ).

tff(fact_1157_one__neq__zero,axiom,
    one_one_real != zero_zero_real ).

tff(fact_1158_one__neq__zero,axiom,
    one_one_int != zero_zero_int ).

tff(fact_1159_right__minus__eq,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba) = zero_zero_rat )
    <=> ( Aa = Ba ) ) ).

tff(fact_1160_right__minus__eq,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Aa),Ba) = zero_zero_complex )
    <=> ( Aa = Ba ) ) ).

tff(fact_1161_right__minus__eq,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba) = zero_zero_real )
    <=> ( Aa = Ba ) ) ).

tff(fact_1162_right__minus__eq,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = zero_zero_int )
    <=> ( Aa = Ba ) ) ).

tff(fact_1163_eq__iff__diff__eq__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = Ba )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba) = zero_zero_rat ) ) ).

tff(fact_1164_eq__iff__diff__eq__0,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( Aa = Ba )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Aa),Ba) = zero_zero_complex ) ) ).

tff(fact_1165_eq__iff__diff__eq__0,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = Ba )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba) = zero_zero_real ) ) ).

tff(fact_1166_eq__iff__diff__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = Ba )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = zero_zero_int ) ) ).

tff(fact_1167_diff__self,axiom,
    ! [A_169: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_169),A_169) = zero_zero_rat ).

tff(fact_1168_diff__self,axiom,
    ! [A_169: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_169),A_169) = zero_zero_complex ).

tff(fact_1169_diff__self,axiom,
    ! [A_169: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_169),A_169) = zero_zero_real ).

tff(fact_1170_diff__self,axiom,
    ! [A_169: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_169),A_169) = zero_zero_int ).

tff(fact_1171_diff__0__right,axiom,
    ! [A_168: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_168),zero_zero_rat) = A_168 ).

tff(fact_1172_diff__0__right,axiom,
    ! [A_168: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_168),zero_zero_complex) = A_168 ).

tff(fact_1173_diff__0__right,axiom,
    ! [A_168: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_168),zero_zero_real) = A_168 ).

tff(fact_1174_diff__0__right,axiom,
    ! [A_168: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_168),zero_zero_int) = A_168 ).

tff(fact_1175_int__0__neq__1,axiom,
    zero_zero_int != one_one_int ).

tff(fact_1176_zadd__0__right,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),zero_zero_int) = Z_1 ).

tff(fact_1177_zadd__0,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1) = Z_1 ).

tff(fact_1178_ceiling__less__cancel,axiom,
    ! [X_35: rat,Y_22: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1419379755ng_rat(X_35)),archim1419379755ng_rat(Y_22)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_35),Y_22)) ) ).

tff(fact_1179_ceiling__less__cancel,axiom,
    ! [X_35: real,Y_22: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_35)),archim856651990g_real(Y_22)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_35),Y_22)) ) ).

tff(fact_1180_ceiling__mono,axiom,
    ! [Y_21: real,X_34: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),X_34))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(Y_21)),archim856651990g_real(X_34))) ) ).

tff(fact_1181_ceiling__mono,axiom,
    ! [Y_21: rat,X_34: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),X_34))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1419379755ng_rat(Y_21)),archim1419379755ng_rat(X_34))) ) ).

tff(fact_1182_ceiling__one,axiom,
    archim1419379755ng_rat(one_one_rat) = one_one_int ).

tff(fact_1183_ceiling__one,axiom,
    archim856651990g_real(one_one_real) = one_one_int ).

tff(fact_1184_ceiling__number__of,axiom,
    ! [V_12: int] : archim856651990g_real(hAPP_int_real(number267125858f_real,V_12)) = hAPP_int_int(number_number_of_int,V_12) ).

tff(fact_1185_ceiling__number__of,axiom,
    ! [V_12: int] : archim1419379755ng_rat(hAPP_int_rat(number_number_of_rat,V_12)) = hAPP_int_int(number_number_of_int,V_12) ).

tff(fact_1186_zero__le__square,axiom,
    ! [A_167: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_167),A_167))) ).

tff(fact_1187_zero__le__square,axiom,
    ! [A_167: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_167),A_167))) ).

tff(fact_1188_zero__le__square,axiom,
    ! [A_167: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_167),A_167))) ).

tff(fact_1189_zero__le__mult__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),Ba)))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Aa))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Ba)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ba),zero_zero_rat)) ) ) ) ).

tff(fact_1190_zero__le__mult__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),Ba)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ba)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),zero_zero_real)) ) ) ) ).

tff(fact_1191_zero__le__mult__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) ) ) ) ).

tff(fact_1192_mult__le__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),Ba)),zero_zero_rat))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Aa))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ba),zero_zero_rat)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Ba)) ) ) ) ).

tff(fact_1193_mult__le__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),Ba)),zero_zero_real))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),zero_zero_real)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ba)) ) ) ) ).

tff(fact_1194_mult__le__0__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba)),zero_zero_int))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) ) ) ) ).

tff(fact_1195_mult__nonneg__nonneg,axiom,
    ! [B_135: rat,A_166: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_166))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_135))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_166),B_135))) ) ) ).

tff(fact_1196_mult__nonneg__nonneg,axiom,
    ! [B_135: quickcheck_code_int,A_166: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_166))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_135))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_166),B_135))) ) ) ).

tff(fact_1197_mult__nonneg__nonneg,axiom,
    ! [B_135: code_code_numeral,A_166: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_166))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_135))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_166),B_135))) ) ) ).

tff(fact_1198_mult__nonneg__nonneg,axiom,
    ! [B_135: nat,A_166: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_166))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_135))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_166),B_135))) ) ) ).

tff(fact_1199_mult__nonneg__nonneg,axiom,
    ! [B_135: real,A_166: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_166))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_135))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_166),B_135))) ) ) ).

tff(fact_1200_mult__nonneg__nonneg,axiom,
    ! [B_135: int,A_166: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_166))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_135))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_166),B_135))) ) ) ).

tff(fact_1201_mult__nonneg__nonpos,axiom,
    ! [B_134: rat,A_165: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_165))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_134),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_165),B_134)),zero_zero_rat)) ) ) ).

tff(fact_1202_mult__nonneg__nonpos,axiom,
    ! [B_134: quickcheck_code_int,A_165: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_165))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_134),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_165),B_134)),zero_z891286103de_int)) ) ) ).

tff(fact_1203_mult__nonneg__nonpos,axiom,
    ! [B_134: code_code_numeral,A_165: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_165))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_134),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_165),B_134)),zero_z126310315umeral)) ) ) ).

tff(fact_1204_mult__nonneg__nonpos,axiom,
    ! [B_134: nat,A_165: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_165))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_134),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_165),B_134)),zero_zero_nat)) ) ) ).

tff(fact_1205_mult__nonneg__nonpos,axiom,
    ! [B_134: real,A_165: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_165))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_134),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_165),B_134)),zero_zero_real)) ) ) ).

tff(fact_1206_mult__nonneg__nonpos,axiom,
    ! [B_134: int,A_165: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_165))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_134),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_165),B_134)),zero_zero_int)) ) ) ).

tff(fact_1207_mult__nonneg__nonpos2,axiom,
    ! [B_133: rat,A_164: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_164))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_133),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_133),A_164)),zero_zero_rat)) ) ) ).

tff(fact_1208_mult__nonneg__nonpos2,axiom,
    ! [B_133: quickcheck_code_int,A_164: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_164))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_133),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_133),A_164)),zero_z891286103de_int)) ) ) ).

tff(fact_1209_mult__nonneg__nonpos2,axiom,
    ! [B_133: code_code_numeral,A_164: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_164))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_133),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_133),A_164)),zero_z126310315umeral)) ) ) ).

tff(fact_1210_mult__nonneg__nonpos2,axiom,
    ! [B_133: nat,A_164: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_164))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_133),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_133),A_164)),zero_zero_nat)) ) ) ).

tff(fact_1211_mult__nonneg__nonpos2,axiom,
    ! [B_133: real,A_164: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_164))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_133),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_133),A_164)),zero_zero_real)) ) ) ).

tff(fact_1212_mult__nonneg__nonpos2,axiom,
    ! [B_133: int,A_164: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_164))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_133),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_133),A_164)),zero_zero_int)) ) ) ).

tff(fact_1213_mult__nonpos__nonneg,axiom,
    ! [B_132: rat,A_163: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_163),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_132))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_163),B_132)),zero_zero_rat)) ) ) ).

tff(fact_1214_mult__nonpos__nonneg,axiom,
    ! [B_132: quickcheck_code_int,A_163: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_163),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_132))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_163),B_132)),zero_z891286103de_int)) ) ) ).

tff(fact_1215_mult__nonpos__nonneg,axiom,
    ! [B_132: code_code_numeral,A_163: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_163),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_132))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_163),B_132)),zero_z126310315umeral)) ) ) ).

tff(fact_1216_mult__nonpos__nonneg,axiom,
    ! [B_132: nat,A_163: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_163),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_132))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_163),B_132)),zero_zero_nat)) ) ) ).

tff(fact_1217_mult__nonpos__nonneg,axiom,
    ! [B_132: real,A_163: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_163),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_132))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_163),B_132)),zero_zero_real)) ) ) ).

tff(fact_1218_mult__nonpos__nonneg,axiom,
    ! [B_132: int,A_163: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_163),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_132))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_163),B_132)),zero_zero_int)) ) ) ).

tff(fact_1219_mult__nonpos__nonpos,axiom,
    ! [B_131: rat,A_162: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_162),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_131),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_162),B_131))) ) ) ).

tff(fact_1220_mult__nonpos__nonpos,axiom,
    ! [B_131: real,A_162: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_162),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_131),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_162),B_131))) ) ) ).

tff(fact_1221_mult__nonpos__nonpos,axiom,
    ! [B_131: int,A_162: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_162),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_131),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_162),B_131))) ) ) ).

tff(fact_1222_mult__right__mono,axiom,
    ! [C_63: rat,A_161: rat,B_130: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_161),B_130))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_63))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_161),C_63)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_130),C_63))) ) ) ).

tff(fact_1223_mult__right__mono,axiom,
    ! [C_63: quickcheck_code_int,A_161: quickcheck_code_int,B_130: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_161),B_130))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_63))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_161),C_63)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_130),C_63))) ) ) ).

tff(fact_1224_mult__right__mono,axiom,
    ! [C_63: code_code_numeral,A_161: code_code_numeral,B_130: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_161),B_130))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_63))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_161),C_63)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_130),C_63))) ) ) ).

tff(fact_1225_mult__right__mono,axiom,
    ! [C_63: nat,A_161: nat,B_130: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_161),B_130))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_63))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_161),C_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_130),C_63))) ) ) ).

tff(fact_1226_mult__right__mono,axiom,
    ! [C_63: real,A_161: real,B_130: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_161),B_130))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_63))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_161),C_63)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_130),C_63))) ) ) ).

tff(fact_1227_mult__right__mono,axiom,
    ! [C_63: int,A_161: int,B_130: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_161),B_130))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_63))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_161),C_63)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_130),C_63))) ) ) ).

tff(fact_1228_mult__left__mono,axiom,
    ! [C_62: rat,A_160: rat,B_129: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_160),B_129))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_62))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_62),A_160)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_62),B_129))) ) ) ).

tff(fact_1229_mult__left__mono,axiom,
    ! [C_62: quickcheck_code_int,A_160: quickcheck_code_int,B_129: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_160),B_129))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_62))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_62),A_160)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_62),B_129))) ) ) ).

tff(fact_1230_mult__left__mono,axiom,
    ! [C_62: code_code_numeral,A_160: code_code_numeral,B_129: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_160),B_129))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_62))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_62),A_160)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_62),B_129))) ) ) ).

tff(fact_1231_mult__left__mono,axiom,
    ! [C_62: nat,A_160: nat,B_129: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_160),B_129))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_62))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_62),A_160)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_62),B_129))) ) ) ).

tff(fact_1232_mult__left__mono,axiom,
    ! [C_62: real,A_160: real,B_129: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_160),B_129))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_62))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_62),A_160)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_62),B_129))) ) ) ).

tff(fact_1233_mult__left__mono,axiom,
    ! [C_62: int,A_160: int,B_129: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_160),B_129))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_62))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_62),A_160)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_62),B_129))) ) ) ).

tff(fact_1234_comm__mult__left__mono,axiom,
    ! [C_61: rat,A_159: rat,B_128: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_159),B_128))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_61))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_61),A_159)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_61),B_128))) ) ) ).

tff(fact_1235_comm__mult__left__mono,axiom,
    ! [C_61: quickcheck_code_int,A_159: quickcheck_code_int,B_128: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_159),B_128))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_61))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_61),A_159)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_61),B_128))) ) ) ).

tff(fact_1236_comm__mult__left__mono,axiom,
    ! [C_61: code_code_numeral,A_159: code_code_numeral,B_128: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_159),B_128))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_61))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_61),A_159)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_61),B_128))) ) ) ).

tff(fact_1237_comm__mult__left__mono,axiom,
    ! [C_61: nat,A_159: nat,B_128: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_159),B_128))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_61))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_61),A_159)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_61),B_128))) ) ) ).

tff(fact_1238_comm__mult__left__mono,axiom,
    ! [C_61: real,A_159: real,B_128: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_159),B_128))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_61))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_61),A_159)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_61),B_128))) ) ) ).

tff(fact_1239_comm__mult__left__mono,axiom,
    ! [C_61: int,A_159: int,B_128: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_159),B_128))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_61))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_61),A_159)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_61),B_128))) ) ) ).

tff(fact_1240_mult__right__mono__neg,axiom,
    ! [C_60: rat,B_127: rat,A_158: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_127),A_158))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_60),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_158),C_60)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_127),C_60))) ) ) ).

tff(fact_1241_mult__right__mono__neg,axiom,
    ! [C_60: real,B_127: real,A_158: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_127),A_158))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_60),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_158),C_60)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_127),C_60))) ) ) ).

tff(fact_1242_mult__right__mono__neg,axiom,
    ! [C_60: int,B_127: int,A_158: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_127),A_158))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_60),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_158),C_60)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_127),C_60))) ) ) ).

tff(fact_1243_mult__left__mono__neg,axiom,
    ! [C_59: rat,B_126: rat,A_157: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_126),A_157))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_59),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_59),A_157)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_59),B_126))) ) ) ).

tff(fact_1244_mult__left__mono__neg,axiom,
    ! [C_59: real,B_126: real,A_157: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_126),A_157))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_59),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_59),A_157)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_59),B_126))) ) ) ).

tff(fact_1245_mult__left__mono__neg,axiom,
    ! [C_59: int,B_126: int,A_157: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_126),A_157))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_59),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_59),A_157)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_59),B_126))) ) ) ).

tff(fact_1246_mult__mono_H,axiom,
    ! [C_58: rat,D_13: rat,A_156: rat,B_125: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_156),B_125))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_58),D_13))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_156))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_58))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_156),C_58)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_125),D_13))) ) ) ) ) ).

tff(fact_1247_mult__mono_H,axiom,
    ! [C_58: quickcheck_code_int,D_13: quickcheck_code_int,A_156: quickcheck_code_int,B_125: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_156),B_125))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_58),D_13))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_156))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_58))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_156),C_58)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_125),D_13))) ) ) ) ) ).

tff(fact_1248_mult__mono_H,axiom,
    ! [C_58: code_code_numeral,D_13: code_code_numeral,A_156: code_code_numeral,B_125: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_156),B_125))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_58),D_13))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_156))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_58))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_156),C_58)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_125),D_13))) ) ) ) ) ).

tff(fact_1249_mult__mono_H,axiom,
    ! [C_58: nat,D_13: nat,A_156: nat,B_125: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_156),B_125))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_58),D_13))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_156))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_58))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_156),C_58)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_125),D_13))) ) ) ) ) ).

tff(fact_1250_mult__mono_H,axiom,
    ! [C_58: real,D_13: real,A_156: real,B_125: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_156),B_125))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_58),D_13))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_156))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_58))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_156),C_58)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_125),D_13))) ) ) ) ) ).

tff(fact_1251_mult__mono_H,axiom,
    ! [C_58: int,D_13: int,A_156: int,B_125: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_156),B_125))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_58),D_13))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_156))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_58))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_156),C_58)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_125),D_13))) ) ) ) ) ).

tff(fact_1252_mult__mono,axiom,
    ! [C_57: rat,D_12: rat,A_155: rat,B_124: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_155),B_124))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_57),D_12))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_124))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_57))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_155),C_57)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_124),D_12))) ) ) ) ) ).

tff(fact_1253_mult__mono,axiom,
    ! [C_57: quickcheck_code_int,D_12: quickcheck_code_int,A_155: quickcheck_code_int,B_124: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_155),B_124))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_57),D_12))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_124))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_57))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_155),C_57)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_124),D_12))) ) ) ) ) ).

tff(fact_1254_mult__mono,axiom,
    ! [C_57: code_code_numeral,D_12: code_code_numeral,A_155: code_code_numeral,B_124: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_155),B_124))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_57),D_12))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_124))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_57))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_155),C_57)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_124),D_12))) ) ) ) ) ).

tff(fact_1255_mult__mono,axiom,
    ! [C_57: nat,D_12: nat,A_155: nat,B_124: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_155),B_124))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_57),D_12))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_124))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_57))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_155),C_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_124),D_12))) ) ) ) ) ).

tff(fact_1256_mult__mono,axiom,
    ! [C_57: real,D_12: real,A_155: real,B_124: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_155),B_124))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_57),D_12))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_124))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_57))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_155),C_57)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_124),D_12))) ) ) ) ) ).

tff(fact_1257_mult__mono,axiom,
    ! [C_57: int,D_12: int,A_155: int,B_124: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_155),B_124))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_57),D_12))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_124))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_57))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_155),C_57)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_124),D_12))) ) ) ) ) ).

tff(fact_1258_split__mult__pos__le,axiom,
    ! [B_123: rat,A_154: rat] :
      ( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_154))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_123)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_154),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_123),zero_zero_rat)) ) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_154),B_123))) ) ).

tff(fact_1259_split__mult__pos__le,axiom,
    ! [B_123: real,A_154: real] :
      ( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_154))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_123)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_154),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_123),zero_zero_real)) ) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_154),B_123))) ) ).

tff(fact_1260_split__mult__pos__le,axiom,
    ! [B_123: int,A_154: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_154))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_123)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_154),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_123),zero_zero_int)) ) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_154),B_123))) ) ).

tff(fact_1261_split__mult__neg__le,axiom,
    ! [B_122: rat,A_153: rat] :
      ( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_153))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_122),zero_zero_rat)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_153),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_122)) ) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_153),B_122)),zero_zero_rat)) ) ).

tff(fact_1262_split__mult__neg__le,axiom,
    ! [B_122: quickcheck_code_int,A_153: quickcheck_code_int] :
      ( ( ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_153))
          & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_122),zero_z891286103de_int)) )
        | ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_153),zero_z891286103de_int))
          & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_122)) ) )
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_153),B_122)),zero_z891286103de_int)) ) ).

tff(fact_1263_split__mult__neg__le,axiom,
    ! [B_122: code_code_numeral,A_153: code_code_numeral] :
      ( ( ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_153))
          & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_122),zero_z126310315umeral)) )
        | ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_153),zero_z126310315umeral))
          & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_122)) ) )
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_153),B_122)),zero_z126310315umeral)) ) ).

tff(fact_1264_split__mult__neg__le,axiom,
    ! [B_122: nat,A_153: nat] :
      ( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_153))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_122),zero_zero_nat)) )
        | ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_153),zero_zero_nat))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_122)) ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_153),B_122)),zero_zero_nat)) ) ).

tff(fact_1265_split__mult__neg__le,axiom,
    ! [B_122: real,A_153: real] :
      ( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_153))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_122),zero_zero_real)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_153),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_122)) ) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_153),B_122)),zero_zero_real)) ) ).

tff(fact_1266_split__mult__neg__le,axiom,
    ! [B_122: int,A_153: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_153))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_122),zero_zero_int)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_153),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_122)) ) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_153),B_122)),zero_zero_int)) ) ).

tff(fact_1267_not__square__less__zero,axiom,
    ! [A_152: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_152),A_152)),zero_zero_rat)) ).

tff(fact_1268_not__square__less__zero,axiom,
    ! [A_152: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_152),A_152)),zero_zero_real)) ).

tff(fact_1269_not__square__less__zero,axiom,
    ! [A_152: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_152),A_152)),zero_zero_int)) ).

tff(fact_1270_mult__less__cancel__right__disj,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),C)))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ba),Aa)) ) ) ) ).

tff(fact_1271_mult__less__cancel__right__disj,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),C)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ) ).

tff(fact_1272_mult__less__cancel__right__disj,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),C)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ) ).

tff(fact_1273_mult__less__cancel__left__disj,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Ba)))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ba),Aa)) ) ) ) ).

tff(fact_1274_mult__less__cancel__left__disj,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Ba)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ) ).

tff(fact_1275_mult__less__cancel__left__disj,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ) ).

tff(fact_1276_mult__less__cancel__left__pos,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Ba)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba)) ) ) ).

tff(fact_1277_mult__less__cancel__left__pos,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Ba)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ) ).

tff(fact_1278_mult__less__cancel__left__pos,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ) ).

tff(fact_1279_mult__pos__pos,axiom,
    ! [B_121: rat,A_151: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_151))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_121))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_151),B_121))) ) ) ).

tff(fact_1280_mult__pos__pos,axiom,
    ! [B_121: quickcheck_code_int,A_151: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_151))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_121))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_151),B_121))) ) ) ).

tff(fact_1281_mult__pos__pos,axiom,
    ! [B_121: code_code_numeral,A_151: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_151))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_121))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_151),B_121))) ) ) ).

tff(fact_1282_mult__pos__pos,axiom,
    ! [B_121: nat,A_151: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_151))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_121))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_151),B_121))) ) ) ).

tff(fact_1283_mult__pos__pos,axiom,
    ! [B_121: real,A_151: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_151))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_121))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_151),B_121))) ) ) ).

tff(fact_1284_mult__pos__pos,axiom,
    ! [B_121: int,A_151: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_151))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_121))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_151),B_121))) ) ) ).

tff(fact_1285_mult__pos__neg,axiom,
    ! [B_120: rat,A_150: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_150))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_120),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_150),B_120)),zero_zero_rat)) ) ) ).

tff(fact_1286_mult__pos__neg,axiom,
    ! [B_120: quickcheck_code_int,A_150: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_150))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_120),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_150),B_120)),zero_z891286103de_int)) ) ) ).

tff(fact_1287_mult__pos__neg,axiom,
    ! [B_120: code_code_numeral,A_150: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_150))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_120),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_150),B_120)),zero_z126310315umeral)) ) ) ).

tff(fact_1288_mult__pos__neg,axiom,
    ! [B_120: nat,A_150: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_150))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_120),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_150),B_120)),zero_zero_nat)) ) ) ).

tff(fact_1289_mult__pos__neg,axiom,
    ! [B_120: real,A_150: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_150))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_120),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_150),B_120)),zero_zero_real)) ) ) ).

tff(fact_1290_mult__pos__neg,axiom,
    ! [B_120: int,A_150: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_150))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_120),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_150),B_120)),zero_zero_int)) ) ) ).

tff(fact_1291_mult__pos__neg2,axiom,
    ! [B_119: rat,A_149: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_149))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_119),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_119),A_149)),zero_zero_rat)) ) ) ).

tff(fact_1292_mult__pos__neg2,axiom,
    ! [B_119: quickcheck_code_int,A_149: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_149))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_119),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_119),A_149)),zero_z891286103de_int)) ) ) ).

tff(fact_1293_mult__pos__neg2,axiom,
    ! [B_119: code_code_numeral,A_149: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_149))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_119),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_119),A_149)),zero_z126310315umeral)) ) ) ).

tff(fact_1294_mult__pos__neg2,axiom,
    ! [B_119: nat,A_149: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_149))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_119),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_119),A_149)),zero_zero_nat)) ) ) ).

tff(fact_1295_mult__pos__neg2,axiom,
    ! [B_119: real,A_149: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_149))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_119),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_119),A_149)),zero_zero_real)) ) ) ).

tff(fact_1296_mult__pos__neg2,axiom,
    ! [B_119: int,A_149: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_149))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_119),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_119),A_149)),zero_zero_int)) ) ) ).

tff(fact_1297_zero__less__mult__pos,axiom,
    ! [A_148: rat,B_118: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_148),B_118)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_148))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_118)) ) ) ).

tff(fact_1298_zero__less__mult__pos,axiom,
    ! [A_148: quickcheck_code_int,B_118: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_148),B_118)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_148))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_118)) ) ) ).

tff(fact_1299_zero__less__mult__pos,axiom,
    ! [A_148: code_code_numeral,B_118: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_148),B_118)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_148))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_118)) ) ) ).

tff(fact_1300_zero__less__mult__pos,axiom,
    ! [A_148: nat,B_118: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_148),B_118)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_148))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_118)) ) ) ).

tff(fact_1301_zero__less__mult__pos,axiom,
    ! [A_148: real,B_118: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_148),B_118)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_148))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_118)) ) ) ).

tff(fact_1302_zero__less__mult__pos,axiom,
    ! [A_148: int,B_118: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_148),B_118)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_148))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_118)) ) ) ).

tff(fact_1303_zero__less__mult__pos2,axiom,
    ! [B_117: rat,A_147: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_117),A_147)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_147))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_117)) ) ) ).

tff(fact_1304_zero__less__mult__pos2,axiom,
    ! [B_117: quickcheck_code_int,A_147: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_117),A_147)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_147))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_117)) ) ) ).

tff(fact_1305_zero__less__mult__pos2,axiom,
    ! [B_117: code_code_numeral,A_147: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_117),A_147)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_147))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_117)) ) ) ).

tff(fact_1306_zero__less__mult__pos2,axiom,
    ! [B_117: nat,A_147: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_117),A_147)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_147))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_117)) ) ) ).

tff(fact_1307_zero__less__mult__pos2,axiom,
    ! [B_117: real,A_147: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_117),A_147)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_147))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_117)) ) ) ).

tff(fact_1308_zero__less__mult__pos2,axiom,
    ! [B_117: int,A_147: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_117),A_147)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_147))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_117)) ) ) ).

tff(fact_1309_mult__less__cancel__left__neg,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Ba)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ba),Aa)) ) ) ).

tff(fact_1310_mult__less__cancel__left__neg,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Ba)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ).

tff(fact_1311_mult__less__cancel__left__neg,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ).

tff(fact_1312_mult__neg__pos,axiom,
    ! [B_116: rat,A_146: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_146),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_116))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_146),B_116)),zero_zero_rat)) ) ) ).

tff(fact_1313_mult__neg__pos,axiom,
    ! [B_116: quickcheck_code_int,A_146: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_146),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_146),B_116)),zero_z891286103de_int)) ) ) ).

tff(fact_1314_mult__neg__pos,axiom,
    ! [B_116: code_code_numeral,A_146: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_146),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_146),B_116)),zero_z126310315umeral)) ) ) ).

tff(fact_1315_mult__neg__pos,axiom,
    ! [B_116: nat,A_146: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_146),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_146),B_116)),zero_zero_nat)) ) ) ).

tff(fact_1316_mult__neg__pos,axiom,
    ! [B_116: real,A_146: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_146),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_116))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_146),B_116)),zero_zero_real)) ) ) ).

tff(fact_1317_mult__neg__pos,axiom,
    ! [B_116: int,A_146: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_146),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_146),B_116)),zero_zero_int)) ) ) ).

tff(fact_1318_mult__neg__neg,axiom,
    ! [B_115: rat,A_145: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_145),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_145),B_115))) ) ) ).

tff(fact_1319_mult__neg__neg,axiom,
    ! [B_115: real,A_145: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_145),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_145),B_115))) ) ) ).

tff(fact_1320_mult__neg__neg,axiom,
    ! [B_115: int,A_145: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_145),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_145),B_115))) ) ) ).

tff(fact_1321_mult__strict__right__mono,axiom,
    ! [C_56: rat,A_144: rat,B_114: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_144),B_114))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_56))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_144),C_56)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_114),C_56))) ) ) ).

tff(fact_1322_mult__strict__right__mono,axiom,
    ! [C_56: quickcheck_code_int,A_144: quickcheck_code_int,B_114: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_144),B_114))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_56))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_144),C_56)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_114),C_56))) ) ) ).

tff(fact_1323_mult__strict__right__mono,axiom,
    ! [C_56: code_code_numeral,A_144: code_code_numeral,B_114: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_144),B_114))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_56))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_144),C_56)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_114),C_56))) ) ) ).

tff(fact_1324_mult__strict__right__mono,axiom,
    ! [C_56: nat,A_144: nat,B_114: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_144),B_114))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_56))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_144),C_56)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_114),C_56))) ) ) ).

tff(fact_1325_mult__strict__right__mono,axiom,
    ! [C_56: real,A_144: real,B_114: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_144),B_114))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_56))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_144),C_56)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_114),C_56))) ) ) ).

tff(fact_1326_mult__strict__right__mono,axiom,
    ! [C_56: int,A_144: int,B_114: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_144),B_114))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_56))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_144),C_56)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_114),C_56))) ) ) ).

tff(fact_1327_mult__strict__left__mono,axiom,
    ! [C_55: rat,A_143: rat,B_113: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_143),B_113))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_55))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_55),A_143)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_55),B_113))) ) ) ).

tff(fact_1328_mult__strict__left__mono,axiom,
    ! [C_55: quickcheck_code_int,A_143: quickcheck_code_int,B_113: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_143),B_113))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_55))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_55),A_143)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_55),B_113))) ) ) ).

tff(fact_1329_mult__strict__left__mono,axiom,
    ! [C_55: code_code_numeral,A_143: code_code_numeral,B_113: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_143),B_113))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_55))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_55),A_143)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_55),B_113))) ) ) ).

tff(fact_1330_mult__strict__left__mono,axiom,
    ! [C_55: nat,A_143: nat,B_113: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_143),B_113))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_55))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_55),A_143)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_55),B_113))) ) ) ).

tff(fact_1331_mult__strict__left__mono,axiom,
    ! [C_55: real,A_143: real,B_113: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_143),B_113))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_55))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_55),A_143)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_55),B_113))) ) ) ).

tff(fact_1332_mult__strict__left__mono,axiom,
    ! [C_55: int,A_143: int,B_113: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_143),B_113))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_55))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_55),A_143)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_55),B_113))) ) ) ).

tff(fact_1333_comm__mult__strict__left__mono,axiom,
    ! [C_54: rat,A_142: rat,B_112: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),B_112))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_54))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_54),A_142)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_54),B_112))) ) ) ).

tff(fact_1334_comm__mult__strict__left__mono,axiom,
    ! [C_54: quickcheck_code_int,A_142: quickcheck_code_int,B_112: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_142),B_112))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_54))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_54),A_142)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_54),B_112))) ) ) ).

tff(fact_1335_comm__mult__strict__left__mono,axiom,
    ! [C_54: code_code_numeral,A_142: code_code_numeral,B_112: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_142),B_112))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_54))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_54),A_142)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_54),B_112))) ) ) ).

tff(fact_1336_comm__mult__strict__left__mono,axiom,
    ! [C_54: nat,A_142: nat,B_112: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_142),B_112))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_54))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_54),A_142)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_54),B_112))) ) ) ).

tff(fact_1337_comm__mult__strict__left__mono,axiom,
    ! [C_54: real,A_142: real,B_112: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),B_112))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_54))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_54),A_142)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_54),B_112))) ) ) ).

tff(fact_1338_comm__mult__strict__left__mono,axiom,
    ! [C_54: int,A_142: int,B_112: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),B_112))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_54))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_54),A_142)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_54),B_112))) ) ) ).

tff(fact_1339_mult__strict__right__mono__neg,axiom,
    ! [C_53: rat,B_111: rat,A_141: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_111),A_141))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_53),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_141),C_53)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_111),C_53))) ) ) ).

tff(fact_1340_mult__strict__right__mono__neg,axiom,
    ! [C_53: real,B_111: real,A_141: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_111),A_141))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_53),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_141),C_53)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_111),C_53))) ) ) ).

tff(fact_1341_mult__strict__right__mono__neg,axiom,
    ! [C_53: int,B_111: int,A_141: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_111),A_141))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_53),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_141),C_53)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_111),C_53))) ) ) ).

tff(fact_1342_mult__strict__left__mono__neg,axiom,
    ! [C_52: rat,B_110: rat,A_140: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_110),A_140))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_52),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_52),A_140)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_52),B_110))) ) ) ).

tff(fact_1343_mult__strict__left__mono__neg,axiom,
    ! [C_52: real,B_110: real,A_140: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_110),A_140))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_52),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_52),A_140)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_52),B_110))) ) ) ).

tff(fact_1344_mult__strict__left__mono__neg,axiom,
    ! [C_52: int,B_110: int,A_140: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_110),A_140))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_52),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_52),A_140)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_52),B_110))) ) ) ).

tff(fact_1345_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Aa)) ) ).

tff(fact_1346_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa)) ) ).

tff(fact_1347_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).

tff(fact_1348_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),zero_zero_rat)) ) ).

tff(fact_1349_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real)) ) ).

tff(fact_1350_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ).

tff(fact_1351_add__nonneg__nonneg,axiom,
    ! [B_109: rat,A_139: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_139))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_109))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_139),B_109))) ) ) ).

tff(fact_1352_add__nonneg__nonneg,axiom,
    ! [B_109: quickcheck_code_int,A_139: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_139))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_109))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_139),B_109))) ) ) ).

tff(fact_1353_add__nonneg__nonneg,axiom,
    ! [B_109: nat,A_139: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_139))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_109))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_139),B_109))) ) ) ).

tff(fact_1354_add__nonneg__nonneg,axiom,
    ! [B_109: real,A_139: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_139))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_109))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_139),B_109))) ) ) ).

tff(fact_1355_add__nonneg__nonneg,axiom,
    ! [B_109: int,A_139: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_139))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_109))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_139),B_109))) ) ) ).

tff(fact_1356_add__nonneg__nonneg,axiom,
    ! [B_109: code_code_numeral,A_139: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_139))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_109))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_139),B_109))) ) ) ).

tff(fact_1357_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: rat,X_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_1))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_2))
       => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Y_2) = zero_zero_rat )
        <=> ( ( X_1 = zero_zero_rat )
            & ( Y_2 = zero_zero_rat ) ) ) ) ) ).

tff(fact_1358_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: quickcheck_code_int,X_1: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_1))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_2))
       => ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_1),Y_2) = zero_z891286103de_int )
        <=> ( ( X_1 = zero_z891286103de_int )
            & ( Y_2 = zero_z891286103de_int ) ) ) ) ) ).

tff(fact_1359_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_2))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2) = zero_zero_nat )
        <=> ( ( X_1 = zero_zero_nat )
            & ( Y_2 = zero_zero_nat ) ) ) ) ) ).

tff(fact_1360_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2))
       => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real )
        <=> ( ( X_1 = zero_zero_real )
            & ( Y_2 = zero_zero_real ) ) ) ) ) ).

tff(fact_1361_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2) = zero_zero_int )
        <=> ( ( X_1 = zero_zero_int )
            & ( Y_2 = zero_zero_int ) ) ) ) ) ).

tff(fact_1362_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: code_code_numeral,X_1: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_1))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_2))
       => ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_1),Y_2) = zero_z126310315umeral )
        <=> ( ( X_1 = zero_z126310315umeral )
            & ( Y_2 = zero_z126310315umeral ) ) ) ) ) ).

tff(fact_1363_add__increasing,axiom,
    ! [B_108: rat,C_51: rat,A_138: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_138))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_108),C_51))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_108),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_138),C_51))) ) ) ).

tff(fact_1364_add__increasing,axiom,
    ! [B_108: quickcheck_code_int,C_51: quickcheck_code_int,A_138: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_138))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_108),C_51))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_108),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_138),C_51))) ) ) ).

tff(fact_1365_add__increasing,axiom,
    ! [B_108: nat,C_51: nat,A_138: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_138))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_108),C_51))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_108),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_138),C_51))) ) ) ).

tff(fact_1366_add__increasing,axiom,
    ! [B_108: real,C_51: real,A_138: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_138))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_108),C_51))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_108),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_138),C_51))) ) ) ).

tff(fact_1367_add__increasing,axiom,
    ! [B_108: int,C_51: int,A_138: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_138))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_108),C_51))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_108),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_138),C_51))) ) ) ).

tff(fact_1368_add__increasing,axiom,
    ! [B_108: code_code_numeral,C_51: code_code_numeral,A_138: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_138))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_108),C_51))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_108),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_138),C_51))) ) ) ).

tff(fact_1369_add__increasing2,axiom,
    ! [B_107: rat,A_137: rat,C_50: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_50))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_107),A_137))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_107),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_137),C_50))) ) ) ).

tff(fact_1370_add__increasing2,axiom,
    ! [B_107: quickcheck_code_int,A_137: quickcheck_code_int,C_50: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_50))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_107),A_137))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_107),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_137),C_50))) ) ) ).

tff(fact_1371_add__increasing2,axiom,
    ! [B_107: nat,A_137: nat,C_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_50))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_107),A_137))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_107),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_137),C_50))) ) ) ).

tff(fact_1372_add__increasing2,axiom,
    ! [B_107: real,A_137: real,C_50: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_50))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_107),A_137))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_107),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_137),C_50))) ) ) ).

tff(fact_1373_add__increasing2,axiom,
    ! [B_107: int,A_137: int,C_50: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_50))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_107),A_137))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_107),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_137),C_50))) ) ) ).

tff(fact_1374_add__increasing2,axiom,
    ! [B_107: code_code_numeral,A_137: code_code_numeral,C_50: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_50))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_107),A_137))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_107),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_137),C_50))) ) ) ).

tff(fact_1375_add__nonpos__nonpos,axiom,
    ! [B_106: rat,A_136: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_136),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_106),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_136),B_106)),zero_zero_rat)) ) ) ).

tff(fact_1376_add__nonpos__nonpos,axiom,
    ! [B_106: quickcheck_code_int,A_136: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_136),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_106),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_136),B_106)),zero_z891286103de_int)) ) ) ).

tff(fact_1377_add__nonpos__nonpos,axiom,
    ! [B_106: nat,A_136: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_136),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_106),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_136),B_106)),zero_zero_nat)) ) ) ).

tff(fact_1378_add__nonpos__nonpos,axiom,
    ! [B_106: real,A_136: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_136),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_106),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_136),B_106)),zero_zero_real)) ) ) ).

tff(fact_1379_add__nonpos__nonpos,axiom,
    ! [B_106: int,A_136: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_136),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_106),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_136),B_106)),zero_zero_int)) ) ) ).

tff(fact_1380_add__nonpos__nonpos,axiom,
    ! [B_106: code_code_numeral,A_136: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_136),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_106),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_136),B_106)),zero_z126310315umeral)) ) ) ).

tff(fact_1381_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Aa)) ) ).

tff(fact_1382_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Aa)) ) ).

tff(fact_1383_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).

tff(fact_1384_even__less__0__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),zero_zero_rat)) ) ).

tff(fact_1385_even__less__0__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),zero_zero_real)) ) ).

tff(fact_1386_even__less__0__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).

tff(fact_1387_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),zero_zero_rat)) ) ).

tff(fact_1388_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),zero_zero_real)) ) ).

tff(fact_1389_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).

tff(fact_1390_add__pos__pos,axiom,
    ! [B_105: rat,A_135: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_135))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_105))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_135),B_105))) ) ) ).

tff(fact_1391_add__pos__pos,axiom,
    ! [B_105: quickcheck_code_int,A_135: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_135))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_105))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_135),B_105))) ) ) ).

tff(fact_1392_add__pos__pos,axiom,
    ! [B_105: nat,A_135: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_135))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_105))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_135),B_105))) ) ) ).

tff(fact_1393_add__pos__pos,axiom,
    ! [B_105: real,A_135: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_135))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_105))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_135),B_105))) ) ) ).

tff(fact_1394_add__pos__pos,axiom,
    ! [B_105: int,A_135: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_135))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_105))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_135),B_105))) ) ) ).

tff(fact_1395_add__pos__pos,axiom,
    ! [B_105: code_code_numeral,A_135: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_135))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_105))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_135),B_105))) ) ) ).

tff(fact_1396_pos__add__strict,axiom,
    ! [B_104: rat,C_49: rat,A_134: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_134))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_104),C_49))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_104),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_134),C_49))) ) ) ).

tff(fact_1397_pos__add__strict,axiom,
    ! [B_104: quickcheck_code_int,C_49: quickcheck_code_int,A_134: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_134))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_104),C_49))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_104),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_134),C_49))) ) ) ).

tff(fact_1398_pos__add__strict,axiom,
    ! [B_104: nat,C_49: nat,A_134: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_134))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_104),C_49))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_104),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_134),C_49))) ) ) ).

tff(fact_1399_pos__add__strict,axiom,
    ! [B_104: real,C_49: real,A_134: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_134))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_104),C_49))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_104),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_134),C_49))) ) ) ).

tff(fact_1400_pos__add__strict,axiom,
    ! [B_104: int,C_49: int,A_134: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_134))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_104),C_49))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_104),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_134),C_49))) ) ) ).

tff(fact_1401_pos__add__strict,axiom,
    ! [B_104: code_code_numeral,C_49: code_code_numeral,A_134: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_134))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_104),C_49))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_104),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_134),C_49))) ) ) ).

tff(fact_1402_add__neg__neg,axiom,
    ! [B_103: rat,A_133: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_133),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_103),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_133),B_103)),zero_zero_rat)) ) ) ).

tff(fact_1403_add__neg__neg,axiom,
    ! [B_103: quickcheck_code_int,A_133: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_133),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_103),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_133),B_103)),zero_z891286103de_int)) ) ) ).

tff(fact_1404_add__neg__neg,axiom,
    ! [B_103: nat,A_133: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_133),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_103),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_133),B_103)),zero_zero_nat)) ) ) ).

tff(fact_1405_add__neg__neg,axiom,
    ! [B_103: real,A_133: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_133),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_103),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_133),B_103)),zero_zero_real)) ) ) ).

tff(fact_1406_add__neg__neg,axiom,
    ! [B_103: int,A_133: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_133),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_103),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_133),B_103)),zero_zero_int)) ) ) ).

tff(fact_1407_add__neg__neg,axiom,
    ! [B_103: code_code_numeral,A_133: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_133),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_103),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_133),B_103)),zero_z126310315umeral)) ) ) ).

tff(fact_1408_zero__le__one,axiom,
    hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

tff(fact_1409_zero__le__one,axiom,
    hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),one_on1684967323de_int)) ).

tff(fact_1410_zero__le__one,axiom,
    hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),one_on1645066479umeral)) ).

tff(fact_1411_zero__le__one,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

tff(fact_1412_zero__le__one,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

tff(fact_1413_zero__le__one,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

tff(fact_1414_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

tff(fact_1415_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),zero_z891286103de_int)) ).

tff(fact_1416_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),zero_z126310315umeral)) ).

tff(fact_1417_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

tff(fact_1418_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

tff(fact_1419_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

tff(fact_1420_zero__less__one,axiom,
    hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

tff(fact_1421_zero__less__one,axiom,
    hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),one_on1684967323de_int)) ).

tff(fact_1422_zero__less__one,axiom,
    hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),one_on1645066479umeral)) ).

tff(fact_1423_zero__less__one,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

tff(fact_1424_zero__less__one,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

tff(fact_1425_zero__less__one,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

tff(fact_1426_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

tff(fact_1427_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),zero_z891286103de_int)) ).

tff(fact_1428_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),zero_z126310315umeral)) ).

tff(fact_1429_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

tff(fact_1430_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

tff(fact_1431_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).

tff(fact_1432_le__iff__diff__le__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba)),zero_zero_rat)) ) ).

tff(fact_1433_le__iff__diff__le__0,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba)),zero_zero_real)) ) ).

tff(fact_1434_le__iff__diff__le__0,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),zero_zero_int)) ) ).

tff(fact_1435_add__scale__eq__noteq,axiom,
    ! [C_48: rat,D_11: rat,A_132: rat,B_102: rat,R_7: rat] :
      ( ( R_7 != zero_zero_rat )
     => ( ( ( A_132 = B_102 )
          & ( C_48 != D_11 ) )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_132),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_7),C_48)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_102),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_7),D_11)) ) ) ) ).

tff(fact_1436_add__scale__eq__noteq,axiom,
    ! [C_48: complex,D_11: complex,A_132: complex,B_102: complex,R_7: complex] :
      ( ( R_7 != zero_zero_complex )
     => ( ( ( A_132 = B_102 )
          & ( C_48 != D_11 ) )
       => ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_132),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_7),C_48)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_102),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_7),D_11)) ) ) ) ).

tff(fact_1437_add__scale__eq__noteq,axiom,
    ! [C_48: nat,D_11: nat,A_132: nat,B_102: nat,R_7: nat] :
      ( ( R_7 != zero_zero_nat )
     => ( ( ( A_132 = B_102 )
          & ( C_48 != D_11 ) )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_132),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_7),C_48)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_102),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_7),D_11)) ) ) ) ).

tff(fact_1438_add__scale__eq__noteq,axiom,
    ! [C_48: real,D_11: real,A_132: real,B_102: real,R_7: real] :
      ( ( R_7 != zero_zero_real )
     => ( ( ( A_132 = B_102 )
          & ( C_48 != D_11 ) )
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_132),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_7),C_48)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_102),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_7),D_11)) ) ) ) ).

tff(fact_1439_add__scale__eq__noteq,axiom,
    ! [C_48: int,D_11: int,A_132: int,B_102: int,R_7: int] :
      ( ( R_7 != zero_zero_int )
     => ( ( ( A_132 = B_102 )
          & ( C_48 != D_11 ) )
       => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_132),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_7),C_48)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_102),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_7),D_11)) ) ) ) ).

tff(fact_1440_less__iff__diff__less__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba)),zero_zero_rat)) ) ).

tff(fact_1441_less__iff__diff__less__0,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba)),zero_zero_real)) ) ).

tff(fact_1442_less__iff__diff__less__0,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),zero_zero_int)) ) ).

tff(fact_1443_Euler_Oaux1,axiom,
    ! [A_2: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A_2))
       => ( ( X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),one_one_int) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),one_one_int))) ) ) ) ).

tff(fact_1444_int__0__less__1,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

tff(fact_1445_zmult__zless__mono2,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2))) ) ) ).

tff(fact_1446_odd__nonzero,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)),Z_1) != zero_zero_int ).

tff(fact_1447_less__bin__lemma,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)) ) ).

tff(fact_1448_one__less__ceiling,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),archim1419379755ng_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),X_1)) ) ).

tff(fact_1449_one__less__ceiling,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) ) ).

tff(fact_1450_ceiling__le__one,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1419379755ng_rat(X_1)),one_one_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),one_one_rat)) ) ).

tff(fact_1451_ceiling__le__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),one_one_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).

tff(fact_1452_number__of__less__ceiling,axiom,
    ! [V: int,X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,V)),archim1419379755ng_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,V)),X_1)) ) ).

tff(fact_1453_number__of__less__ceiling,axiom,
    ! [V: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,V)),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,V)),X_1)) ) ).

tff(fact_1454_ceiling__le__number__of,axiom,
    ! [X_1: real,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(number267125858f_real,V))) ) ).

tff(fact_1455_ceiling__le__number__of,axiom,
    ! [X_1: rat,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1419379755ng_rat(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),hAPP_int_rat(number_number_of_rat,V))) ) ).

tff(fact_1456_ceiling__add__one,axiom,
    ! [X_33: rat] : archim1419379755ng_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_33),one_one_rat)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1419379755ng_rat(X_33)),one_one_int) ).

tff(fact_1457_ceiling__add__one,axiom,
    ! [X_33: real] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_33),one_one_real)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X_33)),one_one_int) ).

tff(fact_1458_ceiling__add__number__of,axiom,
    ! [X_32: rat,V_11: int] : archim1419379755ng_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_32),hAPP_int_rat(number_number_of_rat,V_11))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1419379755ng_rat(X_32)),hAPP_int_int(number_number_of_int,V_11)) ).

tff(fact_1459_ceiling__add__number__of,axiom,
    ! [X_32: real,V_11: int] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_32),hAPP_int_real(number267125858f_real,V_11))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X_32)),hAPP_int_int(number_number_of_int,V_11)) ).

tff(fact_1460_ceiling__diff__one,axiom,
    ! [X_31: rat] : archim1419379755ng_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_31),one_one_rat)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1419379755ng_rat(X_31)),one_one_int) ).

tff(fact_1461_ceiling__diff__one,axiom,
    ! [X_31: real] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_31),one_one_real)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X_31)),one_one_int) ).

tff(fact_1462_ceiling__diff__number__of,axiom,
    ! [X_30: rat,V_10: int] : archim1419379755ng_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_30),hAPP_int_rat(number_number_of_rat,V_10))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1419379755ng_rat(X_30)),hAPP_int_int(number_number_of_int,V_10)) ).

tff(fact_1463_ceiling__diff__number__of,axiom,
    ! [X_30: real,V_10: int] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_30),hAPP_int_real(number267125858f_real,V_10))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X_30)),hAPP_int_int(number_number_of_int,V_10)) ).

tff(fact_1464_mult__left__le__imp__le,axiom,
    ! [C_47: rat,A_131: rat,B_101: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_47),A_131)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_47),B_101)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_47))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_131),B_101)) ) ) ).

tff(fact_1465_mult__left__le__imp__le,axiom,
    ! [C_47: quickcheck_code_int,A_131: quickcheck_code_int,B_101: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_47),A_131)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_47),B_101)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_47))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_131),B_101)) ) ) ).

tff(fact_1466_mult__left__le__imp__le,axiom,
    ! [C_47: code_code_numeral,A_131: code_code_numeral,B_101: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_47),A_131)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_47),B_101)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_47))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_131),B_101)) ) ) ).

tff(fact_1467_mult__left__le__imp__le,axiom,
    ! [C_47: nat,A_131: nat,B_101: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_47),A_131)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_47),B_101)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_47))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_131),B_101)) ) ) ).

tff(fact_1468_mult__left__le__imp__le,axiom,
    ! [C_47: real,A_131: real,B_101: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_47),A_131)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_47),B_101)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_47))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_131),B_101)) ) ) ).

tff(fact_1469_mult__left__le__imp__le,axiom,
    ! [C_47: int,A_131: int,B_101: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_47),A_131)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_47),B_101)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_47))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_131),B_101)) ) ) ).

tff(fact_1470_mult__right__le__imp__le,axiom,
    ! [A_130: rat,C_46: rat,B_100: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),C_46)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_100),C_46)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_46))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_130),B_100)) ) ) ).

tff(fact_1471_mult__right__le__imp__le,axiom,
    ! [A_130: quickcheck_code_int,C_46: quickcheck_code_int,B_100: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),C_46)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_100),C_46)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_46))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_130),B_100)) ) ) ).

tff(fact_1472_mult__right__le__imp__le,axiom,
    ! [A_130: code_code_numeral,C_46: code_code_numeral,B_100: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_130),C_46)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_100),C_46)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_46))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_130),B_100)) ) ) ).

tff(fact_1473_mult__right__le__imp__le,axiom,
    ! [A_130: nat,C_46: nat,B_100: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),C_46)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_100),C_46)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_46))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_130),B_100)) ) ) ).

tff(fact_1474_mult__right__le__imp__le,axiom,
    ! [A_130: real,C_46: real,B_100: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),C_46)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_100),C_46)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_46))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_130),B_100)) ) ) ).

tff(fact_1475_mult__right__le__imp__le,axiom,
    ! [A_130: int,C_46: int,B_100: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),C_46)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_100),C_46)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_46))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_130),B_100)) ) ) ).

tff(fact_1476_mult__less__imp__less__left,axiom,
    ! [C_45: rat,A_129: rat,B_99: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_45),A_129)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_45),B_99)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_45))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_129),B_99)) ) ) ).

tff(fact_1477_mult__less__imp__less__left,axiom,
    ! [C_45: quickcheck_code_int,A_129: quickcheck_code_int,B_99: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_45),A_129)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_45),B_99)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_45))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_129),B_99)) ) ) ).

tff(fact_1478_mult__less__imp__less__left,axiom,
    ! [C_45: code_code_numeral,A_129: code_code_numeral,B_99: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_45),A_129)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_45),B_99)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_45))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_129),B_99)) ) ) ).

tff(fact_1479_mult__less__imp__less__left,axiom,
    ! [C_45: nat,A_129: nat,B_99: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_45),A_129)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_45),B_99)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_45))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_129),B_99)) ) ) ).

tff(fact_1480_mult__less__imp__less__left,axiom,
    ! [C_45: real,A_129: real,B_99: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_45),A_129)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_45),B_99)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_45))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_129),B_99)) ) ) ).

tff(fact_1481_mult__less__imp__less__left,axiom,
    ! [C_45: int,A_129: int,B_99: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_45),A_129)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_45),B_99)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_45))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_129),B_99)) ) ) ).

tff(fact_1482_mult__left__less__imp__less,axiom,
    ! [C_44: rat,A_128: rat,B_98: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_44),A_128)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_44),B_98)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_44))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_128),B_98)) ) ) ).

tff(fact_1483_mult__left__less__imp__less,axiom,
    ! [C_44: quickcheck_code_int,A_128: quickcheck_code_int,B_98: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_44),A_128)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_44),B_98)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_44))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_128),B_98)) ) ) ).

tff(fact_1484_mult__left__less__imp__less,axiom,
    ! [C_44: code_code_numeral,A_128: code_code_numeral,B_98: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_44),A_128)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_44),B_98)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_44))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_128),B_98)) ) ) ).

tff(fact_1485_mult__left__less__imp__less,axiom,
    ! [C_44: nat,A_128: nat,B_98: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_44),A_128)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_44),B_98)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_44))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_128),B_98)) ) ) ).

tff(fact_1486_mult__left__less__imp__less,axiom,
    ! [C_44: real,A_128: real,B_98: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_44),A_128)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_44),B_98)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_44))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_128),B_98)) ) ) ).

tff(fact_1487_mult__left__less__imp__less,axiom,
    ! [C_44: int,A_128: int,B_98: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_44),A_128)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_44),B_98)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_44))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_128),B_98)) ) ) ).

tff(fact_1488_mult__less__imp__less__right,axiom,
    ! [A_127: rat,C_43: rat,B_97: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_127),C_43)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_97),C_43)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_43))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_127),B_97)) ) ) ).

tff(fact_1489_mult__less__imp__less__right,axiom,
    ! [A_127: quickcheck_code_int,C_43: quickcheck_code_int,B_97: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_127),C_43)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_97),C_43)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_43))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_127),B_97)) ) ) ).

tff(fact_1490_mult__less__imp__less__right,axiom,
    ! [A_127: code_code_numeral,C_43: code_code_numeral,B_97: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_127),C_43)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_97),C_43)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_43))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_127),B_97)) ) ) ).

tff(fact_1491_mult__less__imp__less__right,axiom,
    ! [A_127: nat,C_43: nat,B_97: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_127),C_43)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_97),C_43)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_43))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_127),B_97)) ) ) ).

tff(fact_1492_mult__less__imp__less__right,axiom,
    ! [A_127: real,C_43: real,B_97: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_127),C_43)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_97),C_43)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_43))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_127),B_97)) ) ) ).

tff(fact_1493_mult__less__imp__less__right,axiom,
    ! [A_127: int,C_43: int,B_97: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_127),C_43)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_97),C_43)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_43))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_127),B_97)) ) ) ).

tff(fact_1494_mult__right__less__imp__less,axiom,
    ! [A_126: rat,C_42: rat,B_96: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_126),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_96),C_42)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_42))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_126),B_96)) ) ) ).

tff(fact_1495_mult__right__less__imp__less,axiom,
    ! [A_126: quickcheck_code_int,C_42: quickcheck_code_int,B_96: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_126),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_96),C_42)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_42))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_126),B_96)) ) ) ).

tff(fact_1496_mult__right__less__imp__less,axiom,
    ! [A_126: code_code_numeral,C_42: code_code_numeral,B_96: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_126),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_96),C_42)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_42))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_126),B_96)) ) ) ).

tff(fact_1497_mult__right__less__imp__less,axiom,
    ! [A_126: nat,C_42: nat,B_96: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_126),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_96),C_42)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_42))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_126),B_96)) ) ) ).

tff(fact_1498_mult__right__less__imp__less,axiom,
    ! [A_126: real,C_42: real,B_96: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_126),C_42)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_96),C_42)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_42))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_126),B_96)) ) ) ).

tff(fact_1499_mult__right__less__imp__less,axiom,
    ! [A_126: int,C_42: int,B_96: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_126),C_42)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_96),C_42)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_42))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_126),B_96)) ) ) ).

tff(fact_1500_mult__le__less__imp__less,axiom,
    ! [C_41: rat,D_10: rat,A_125: rat,B_95: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_125),B_95))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_41),D_10))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_125))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_41))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_125),C_41)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_95),D_10))) ) ) ) ) ).

tff(fact_1501_mult__le__less__imp__less,axiom,
    ! [C_41: quickcheck_code_int,D_10: quickcheck_code_int,A_125: quickcheck_code_int,B_95: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_125),B_95))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_41),D_10))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_125))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_41))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_125),C_41)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_95),D_10))) ) ) ) ) ).

tff(fact_1502_mult__le__less__imp__less,axiom,
    ! [C_41: code_code_numeral,D_10: code_code_numeral,A_125: code_code_numeral,B_95: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_125),B_95))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_41),D_10))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_125))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_41))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_125),C_41)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_95),D_10))) ) ) ) ) ).

tff(fact_1503_mult__le__less__imp__less,axiom,
    ! [C_41: nat,D_10: nat,A_125: nat,B_95: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_125),B_95))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_41),D_10))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_125))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_41))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_125),C_41)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_95),D_10))) ) ) ) ) ).

tff(fact_1504_mult__le__less__imp__less,axiom,
    ! [C_41: real,D_10: real,A_125: real,B_95: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_125),B_95))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_41),D_10))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_125))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_41))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_125),C_41)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_95),D_10))) ) ) ) ) ).

tff(fact_1505_mult__le__less__imp__less,axiom,
    ! [C_41: int,D_10: int,A_125: int,B_95: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_125),B_95))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_41),D_10))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_125))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_41))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_125),C_41)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_95),D_10))) ) ) ) ) ).

tff(fact_1506_mult__less__le__imp__less,axiom,
    ! [C_40: rat,D_9: rat,A_124: rat,B_94: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_124),B_94))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_40),D_9))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_124))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_40))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_124),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_94),D_9))) ) ) ) ) ).

tff(fact_1507_mult__less__le__imp__less,axiom,
    ! [C_40: quickcheck_code_int,D_9: quickcheck_code_int,A_124: quickcheck_code_int,B_94: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_124),B_94))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_40),D_9))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_124))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_40))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_124),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_94),D_9))) ) ) ) ) ).

tff(fact_1508_mult__less__le__imp__less,axiom,
    ! [C_40: code_code_numeral,D_9: code_code_numeral,A_124: code_code_numeral,B_94: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_124),B_94))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_40),D_9))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_124))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_40))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_124),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_94),D_9))) ) ) ) ) ).

tff(fact_1509_mult__less__le__imp__less,axiom,
    ! [C_40: nat,D_9: nat,A_124: nat,B_94: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_124),B_94))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_40),D_9))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_124))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_40))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_124),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_94),D_9))) ) ) ) ) ).

tff(fact_1510_mult__less__le__imp__less,axiom,
    ! [C_40: real,D_9: real,A_124: real,B_94: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_124),B_94))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_40),D_9))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_124))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_40))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_124),C_40)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_94),D_9))) ) ) ) ) ).

tff(fact_1511_mult__less__le__imp__less,axiom,
    ! [C_40: int,D_9: int,A_124: int,B_94: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_124),B_94))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_40),D_9))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_124))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_40))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_124),C_40)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_94),D_9))) ) ) ) ) ).

tff(fact_1512_mult__strict__mono_H,axiom,
    ! [C_39: rat,D_8: rat,A_123: rat,B_93: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_123),B_93))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_39),D_8))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_123))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_39))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_123),C_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_93),D_8))) ) ) ) ) ).

tff(fact_1513_mult__strict__mono_H,axiom,
    ! [C_39: quickcheck_code_int,D_8: quickcheck_code_int,A_123: quickcheck_code_int,B_93: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_123),B_93))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_39),D_8))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_123))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_39))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_123),C_39)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_93),D_8))) ) ) ) ) ).

tff(fact_1514_mult__strict__mono_H,axiom,
    ! [C_39: code_code_numeral,D_8: code_code_numeral,A_123: code_code_numeral,B_93: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_123),B_93))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_39),D_8))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_123))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_39))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_123),C_39)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_93),D_8))) ) ) ) ) ).

tff(fact_1515_mult__strict__mono_H,axiom,
    ! [C_39: nat,D_8: nat,A_123: nat,B_93: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_123),B_93))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_39),D_8))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_123))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_39))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_123),C_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_93),D_8))) ) ) ) ) ).

tff(fact_1516_mult__strict__mono_H,axiom,
    ! [C_39: real,D_8: real,A_123: real,B_93: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_123),B_93))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_39),D_8))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_123))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_39))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_123),C_39)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_93),D_8))) ) ) ) ) ).

tff(fact_1517_mult__strict__mono_H,axiom,
    ! [C_39: int,D_8: int,A_123: int,B_93: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_123),B_93))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_39),D_8))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_123))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_39))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_123),C_39)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_93),D_8))) ) ) ) ) ).

tff(fact_1518_mult__strict__mono,axiom,
    ! [C_38: rat,D_7: rat,A_122: rat,B_92: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_122),B_92))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_38),D_7))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_92))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_38))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_122),C_38)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_92),D_7))) ) ) ) ) ).

tff(fact_1519_mult__strict__mono,axiom,
    ! [C_38: quickcheck_code_int,D_7: quickcheck_code_int,A_122: quickcheck_code_int,B_92: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_122),B_92))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_38),D_7))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_92))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_38))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_122),C_38)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_92),D_7))) ) ) ) ) ).

tff(fact_1520_mult__strict__mono,axiom,
    ! [C_38: code_code_numeral,D_7: code_code_numeral,A_122: code_code_numeral,B_92: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_122),B_92))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_38),D_7))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_92))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_38))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_122),C_38)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_92),D_7))) ) ) ) ) ).

tff(fact_1521_mult__strict__mono,axiom,
    ! [C_38: nat,D_7: nat,A_122: nat,B_92: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_122),B_92))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_38),D_7))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_92))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_38))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_122),C_38)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_92),D_7))) ) ) ) ) ).

tff(fact_1522_mult__strict__mono,axiom,
    ! [C_38: real,D_7: real,A_122: real,B_92: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_122),B_92))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_38),D_7))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_92))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_38))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_122),C_38)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_92),D_7))) ) ) ) ) ).

tff(fact_1523_mult__strict__mono,axiom,
    ! [C_38: int,D_7: int,A_122: int,B_92: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_122),B_92))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_38),D_7))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_92))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_38))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_122),C_38)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_92),D_7))) ) ) ) ) ).

tff(fact_1524_mult__le__cancel__left__neg,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Ba)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ba),Aa)) ) ) ).

tff(fact_1525_mult__le__cancel__left__neg,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Ba)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),Aa)) ) ) ).

tff(fact_1526_mult__le__cancel__left__neg,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa)) ) ) ).

tff(fact_1527_mult__le__cancel__left__pos,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Aa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),Ba)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba)) ) ) ).

tff(fact_1528_mult__le__cancel__left__pos,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Aa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Ba)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ) ).

tff(fact_1529_mult__le__cancel__left__pos,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Aa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ) ).

tff(fact_1530_add__nonpos__neg,axiom,
    ! [B_91: rat,A_121: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_121),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_91),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_121),B_91)),zero_zero_rat)) ) ) ).

tff(fact_1531_add__nonpos__neg,axiom,
    ! [B_91: quickcheck_code_int,A_121: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_121),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_91),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_121),B_91)),zero_z891286103de_int)) ) ) ).

tff(fact_1532_add__nonpos__neg,axiom,
    ! [B_91: nat,A_121: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_121),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_91),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_121),B_91)),zero_zero_nat)) ) ) ).

tff(fact_1533_add__nonpos__neg,axiom,
    ! [B_91: real,A_121: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_121),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_91),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_121),B_91)),zero_zero_real)) ) ) ).

tff(fact_1534_add__nonpos__neg,axiom,
    ! [B_91: int,A_121: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_121),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_91),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_121),B_91)),zero_zero_int)) ) ) ).

tff(fact_1535_add__nonpos__neg,axiom,
    ! [B_91: code_code_numeral,A_121: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_121),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_91),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_121),B_91)),zero_z126310315umeral)) ) ) ).

tff(fact_1536_add__neg__nonpos,axiom,
    ! [B_90: rat,A_120: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_120),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_90),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_120),B_90)),zero_zero_rat)) ) ) ).

tff(fact_1537_add__neg__nonpos,axiom,
    ! [B_90: quickcheck_code_int,A_120: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_120),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_90),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_120),B_90)),zero_z891286103de_int)) ) ) ).

tff(fact_1538_add__neg__nonpos,axiom,
    ! [B_90: nat,A_120: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_120),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_90),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_120),B_90)),zero_zero_nat)) ) ) ).

tff(fact_1539_add__neg__nonpos,axiom,
    ! [B_90: real,A_120: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_120),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_90),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_120),B_90)),zero_zero_real)) ) ) ).

tff(fact_1540_add__neg__nonpos,axiom,
    ! [B_90: int,A_120: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_120),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_90),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_120),B_90)),zero_zero_int)) ) ) ).

tff(fact_1541_add__neg__nonpos,axiom,
    ! [B_90: code_code_numeral,A_120: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_120),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_90),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_120),B_90)),zero_z126310315umeral)) ) ) ).

tff(fact_1542_add__strict__increasing2,axiom,
    ! [B_89: rat,C_37: rat,A_119: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_119))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_89),C_37))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_89),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_119),C_37))) ) ) ).

tff(fact_1543_add__strict__increasing2,axiom,
    ! [B_89: quickcheck_code_int,C_37: quickcheck_code_int,A_119: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_119))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_89),C_37))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_89),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_119),C_37))) ) ) ).

tff(fact_1544_add__strict__increasing2,axiom,
    ! [B_89: nat,C_37: nat,A_119: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_119))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_89),C_37))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_89),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_119),C_37))) ) ) ).

tff(fact_1545_add__strict__increasing2,axiom,
    ! [B_89: real,C_37: real,A_119: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_119))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_89),C_37))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_89),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_119),C_37))) ) ) ).

tff(fact_1546_add__strict__increasing2,axiom,
    ! [B_89: int,C_37: int,A_119: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_119))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_89),C_37))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_89),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_119),C_37))) ) ) ).

tff(fact_1547_add__strict__increasing2,axiom,
    ! [B_89: code_code_numeral,C_37: code_code_numeral,A_119: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_119))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_89),C_37))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_89),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_119),C_37))) ) ) ).

tff(fact_1548_add__strict__increasing,axiom,
    ! [B_88: rat,C_36: rat,A_118: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_118))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_88),C_36))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_88),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_118),C_36))) ) ) ).

tff(fact_1549_add__strict__increasing,axiom,
    ! [B_88: quickcheck_code_int,C_36: quickcheck_code_int,A_118: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_118))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_88),C_36))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_88),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_118),C_36))) ) ) ).

tff(fact_1550_add__strict__increasing,axiom,
    ! [B_88: nat,C_36: nat,A_118: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_118))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_88),C_36))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_118),C_36))) ) ) ).

tff(fact_1551_add__strict__increasing,axiom,
    ! [B_88: real,C_36: real,A_118: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_118))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_88),C_36))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_88),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_118),C_36))) ) ) ).

tff(fact_1552_add__strict__increasing,axiom,
    ! [B_88: int,C_36: int,A_118: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_118))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_88),C_36))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_88),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_118),C_36))) ) ) ).

tff(fact_1553_add__strict__increasing,axiom,
    ! [B_88: code_code_numeral,C_36: code_code_numeral,A_118: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_118))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_88),C_36))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_88),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_118),C_36))) ) ) ).

tff(fact_1554_add__nonneg__pos,axiom,
    ! [B_87: rat,A_117: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_117))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_87))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_117),B_87))) ) ) ).

tff(fact_1555_add__nonneg__pos,axiom,
    ! [B_87: quickcheck_code_int,A_117: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_117))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_87))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_117),B_87))) ) ) ).

tff(fact_1556_add__nonneg__pos,axiom,
    ! [B_87: nat,A_117: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_117))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_87))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_117),B_87))) ) ) ).

tff(fact_1557_add__nonneg__pos,axiom,
    ! [B_87: real,A_117: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_117))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_87))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_117),B_87))) ) ) ).

tff(fact_1558_add__nonneg__pos,axiom,
    ! [B_87: int,A_117: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_117))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_87))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_117),B_87))) ) ) ).

tff(fact_1559_add__nonneg__pos,axiom,
    ! [B_87: code_code_numeral,A_117: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_117))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_87))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_117),B_87))) ) ) ).

tff(fact_1560_add__pos__nonneg,axiom,
    ! [B_86: rat,A_116: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_116))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_86))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_116),B_86))) ) ) ).

tff(fact_1561_add__pos__nonneg,axiom,
    ! [B_86: quickcheck_code_int,A_116: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_116))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_86))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_116),B_86))) ) ) ).

tff(fact_1562_add__pos__nonneg,axiom,
    ! [B_86: nat,A_116: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_116))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_86))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_116),B_86))) ) ) ).

tff(fact_1563_add__pos__nonneg,axiom,
    ! [B_86: real,A_116: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_116))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_86))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_116),B_86))) ) ) ).

tff(fact_1564_add__pos__nonneg,axiom,
    ! [B_86: int,A_116: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_116))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_86))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_116),B_86))) ) ) ).

tff(fact_1565_add__pos__nonneg,axiom,
    ! [B_86: code_code_numeral,A_116: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_116))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_86))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_116),B_86))) ) ) ).

tff(fact_1566_mult__left__le__one__le,axiom,
    ! [Y_20: rat,X_29: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_29))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_20))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_20),one_one_rat))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_20),X_29)),X_29)) ) ) ) ).

tff(fact_1567_mult__left__le__one__le,axiom,
    ! [Y_20: real,X_29: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_29))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_20))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_20),one_one_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_20),X_29)),X_29)) ) ) ) ).

tff(fact_1568_mult__left__le__one__le,axiom,
    ! [Y_20: int,X_29: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_29))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_20))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_20),one_one_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_20),X_29)),X_29)) ) ) ) ).

tff(fact_1569_mult__right__le__one__le,axiom,
    ! [Y_19: rat,X_28: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_28))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_19))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_19),one_one_rat))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_28),Y_19)),X_28)) ) ) ) ).

tff(fact_1570_mult__right__le__one__le,axiom,
    ! [Y_19: real,X_28: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_28))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_19))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_19),one_one_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_28),Y_19)),X_28)) ) ) ) ).

tff(fact_1571_mult__right__le__one__le,axiom,
    ! [Y_19: int,X_28: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_28))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_19))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_19),one_one_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_28),Y_19)),X_28)) ) ) ) ).

tff(fact_1572_zero__less__two,axiom,
    hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).

tff(fact_1573_zero__less__two,axiom,
    hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int))) ).

tff(fact_1574_zero__less__two,axiom,
    hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral))) ).

tff(fact_1575_zero__less__two,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

tff(fact_1576_zero__less__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real))) ).

tff(fact_1577_zero__less__two,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

tff(fact_1578_int__one__le__iff__zero__less,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).

tff(fact_1579_pos__zmult__eq__1__iff,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

tff(fact_1580_odd__less__0,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).

tff(fact_1581_convex__bound__le,axiom,
    ! [V_9: rat,U_2: rat,Y_18: rat,X_27: rat,A_115: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_27),A_115))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_18),A_115))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_2))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_9))
           => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_9) = one_one_rat )
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_2),X_27)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_18))),A_115)) ) ) ) ) ) ).

tff(fact_1582_convex__bound__le,axiom,
    ! [V_9: real,U_2: real,Y_18: real,X_27: real,A_115: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_27),A_115))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_18),A_115))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_9))
           => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_9) = one_one_real )
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_2),X_27)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_18))),A_115)) ) ) ) ) ) ).

tff(fact_1583_convex__bound__le,axiom,
    ! [V_9: int,U_2: int,Y_18: int,X_27: int,A_115: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),A_115))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_18),A_115))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_9))
           => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_9) = one_one_int )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_2),X_27)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_18))),A_115)) ) ) ) ) ) ).

tff(fact_1584_le__imp__0__less,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1))) ) ).

tff(fact_1585_self__quotient__aux1,axiom,
    ! [R_3: int,Q_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),Q_1)) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),A_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_1)) ) ) ) ).

tff(fact_1586_self__quotient__aux2,axiom,
    ! [R_3: int,Q_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),Q_1)) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),one_one_int)) ) ) ) ).

tff(fact_1587_zdiv__mono2__neg__lemma,axiom,
    ! [B_1: int,Q_1: int,R_3: int,B_4: int,Q_6: int,R_5: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),Q_6)),R_5) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),Q_6)),R_5)),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),B_1))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_6),Q_1)) ) ) ) ) ) ) ).

tff(fact_1588_unique__quotient__lemma__neg,axiom,
    ! [B_1: int,Q_6: int,R_5: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_6)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_3)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_5))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_6)) ) ) ) ) ).

tff(fact_1589_zdiv__mono2__lemma,axiom,
    ! [B_1: int,Q_1: int,R_3: int,B_4: int,Q_6: int,R_5: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),Q_6)),R_5) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),Q_6)),R_5)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_4))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),B_1))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_6)) ) ) ) ) ) ) ).

tff(fact_1590_unique__quotient__lemma,axiom,
    ! [B_1: int,Q_6: int,R_5: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_6)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_3)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_1))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_6),Q_1)) ) ) ) ) ).

tff(fact_1591_q__neg__lemma,axiom,
    ! [B_4: int,Q_6: int,R_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),Q_6)),R_5)),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_6),zero_zero_int)) ) ) ) ).

tff(fact_1592_q__pos__lemma,axiom,
    ! [B_4: int,Q_6: int,R_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),Q_6)),R_5)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_4))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_6)) ) ) ) ).

tff(fact_1593_sum__squares__gt__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))))
    <=> ( ( X_1 != zero_zero_rat )
        | ( Y_2 != zero_zero_rat ) ) ) ).

tff(fact_1594_sum__squares__gt__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))))
    <=> ( ( X_1 != zero_zero_real )
        | ( Y_2 != zero_zero_real ) ) ) ).

tff(fact_1595_sum__squares__gt__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))))
    <=> ( ( X_1 != zero_zero_int )
        | ( Y_2 != zero_zero_int ) ) ) ).

tff(fact_1596_not__sum__squares__lt__zero,axiom,
    ! [X_26: rat,Y_17: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_26),X_26)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_17),Y_17))),zero_zero_rat)) ).

tff(fact_1597_not__sum__squares__lt__zero,axiom,
    ! [X_26: real,Y_17: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_26),X_26)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_17),Y_17))),zero_zero_real)) ).

tff(fact_1598_not__sum__squares__lt__zero,axiom,
    ! [X_26: int,Y_17: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_26),X_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_17),Y_17))),zero_zero_int)) ).

tff(fact_1599_sum__squares__le__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))),zero_zero_rat))
    <=> ( ( X_1 = zero_zero_rat )
        & ( Y_2 = zero_zero_rat ) ) ) ).

tff(fact_1600_sum__squares__le__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))),zero_zero_real))
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_1601_sum__squares__le__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))),zero_zero_int))
    <=> ( ( X_1 = zero_zero_int )
        & ( Y_2 = zero_zero_int ) ) ) ).

tff(fact_1602_sum__squares__eq__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)) = zero_zero_rat )
    <=> ( ( X_1 = zero_zero_rat )
        & ( Y_2 = zero_zero_rat ) ) ) ).

tff(fact_1603_sum__squares__eq__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_1604_sum__squares__eq__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2)) = zero_zero_int )
    <=> ( ( X_1 = zero_zero_int )
        & ( Y_2 = zero_zero_int ) ) ) ).

tff(fact_1605_sum__squares__ge__zero,axiom,
    ! [X_25: rat,Y_16: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_25),X_25)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_16),Y_16)))) ).

tff(fact_1606_sum__squares__ge__zero,axiom,
    ! [X_25: real,Y_16: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_25),X_25)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_16),Y_16)))) ).

tff(fact_1607_sum__squares__ge__zero,axiom,
    ! [X_25: int,Y_16: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_25),X_25)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_16),Y_16)))) ).

tff(fact_1608_xzgcd__linear,axiom,
    ! [M_1: int,R_3: int,S_2: int,T_2: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( ( xzgcd(M_1,N_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_2),T_2)) )
       => ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1)) ) ) ) ).

tff(fact_1609_xzgcda__linear,axiom,
    ! [M_1: int,N_1: int,R_5: int,S_4: int,S_2: int,T_3: int,T_2: int,Rn: int,Sn_1: int,Tn_1: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
     => ( ( xzgcda(M_1,N_1,R_5,R_3,S_4,S_2,T_3,T_2) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Rn),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn_1),Tn_1)) )
       => ( ( R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_4),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_3),N_1)) )
         => ( ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1)) )
           => ( Rn = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Sn_1),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Tn_1),N_1)) ) ) ) ) ) ).

tff(fact_1610_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_1611_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_1612_pos__zmult__pos,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) ) ) ).

tff(fact_1613_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

tff(fact_1614_decr__mult__lemma,axiom,
    ! [K_1: int,P_1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
         => ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(P_1,X_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) ) ) ) ).

tff(fact_1615_incr__mult__lemma,axiom,
    ! [K_1: int,P_1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
         => ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(P_1,X_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) ) ) ) ).

tff(fact_1616_field__le__mult__one__interval,axiom,
    ! [X_24: rat,Y_15: rat] :
      ( ! [Z: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Z))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),one_one_rat))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z),X_24)),Y_15)) ) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_24),Y_15)) ) ).

tff(fact_1617_field__le__mult__one__interval,axiom,
    ! [X_24: real,Y_15: real] :
      ( ! [Z: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),one_one_real))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z),X_24)),Y_15)) ) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_24),Y_15)) ) ).

tff(fact_1618_divmod__int__relI,axiom,
    ! [A_2: int,B_1: int,Q_1: int,R_3: int] :
      ( ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_3) )
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_1)) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_3))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) ) ) )
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3))) ) ) ).

tff(fact_1619_xzgcd__def,axiom,
    ! [M_1: int,N_1: int] : xzgcd(M_1,N_1) = xzgcda(M_1,N_1,M_1,N_1,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).

tff(fact_1620_divmod__int__rel__0,axiom,
    ! [B_1: int] :
      ( ( B_1 != zero_zero_int )
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int))) ) ).

tff(fact_1621_self__remainder,axiom,
    ! [A_2: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,A_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( ( A_2 != zero_zero_int )
       => ( R_3 = zero_zero_int ) ) ) ).

tff(fact_1622_unique__quotient,axiom,
    ! [Q_6: int,R_5: int,A_2: int,B_1: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_6),R_5)))
       => ( ( B_1 != zero_zero_int )
         => ( Q_1 = Q_6 ) ) ) ) ).

tff(fact_1623_unique__remainder,axiom,
    ! [Q_6: int,R_5: int,A_2: int,B_1: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_6),R_5)))
       => ( ( B_1 != zero_zero_int )
         => ( R_3 = R_5 ) ) ) ) ).

tff(fact_1624_self__quotient,axiom,
    ! [A_2: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,A_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( ( A_2 != zero_zero_int )
       => ( Q_1 = one_one_int ) ) ) ).

tff(fact_1625_imp__le__cong,axiom,
    ! [P_2: bool,P_1: bool,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(P_1)
        <=> hBOOL(P_2) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(P_1) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(P_2) ) ) ) ).

tff(fact_1626_conj__le__cong,axiom,
    ! [P_2: bool,P_1: bool,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(P_1)
        <=> hBOOL(P_2) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(P_1) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(P_2) ) ) ) ).

tff(fact_1627_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

tff(fact_1628_field__le__epsilon,axiom,
    ! [X_23: rat,Y_14: rat] :
      ( ! [E: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),E))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_23),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_14),E))) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_23),Y_14)) ) ).

tff(fact_1629_field__le__epsilon,axiom,
    ! [X_23: real,Y_14: real] :
      ( ! [E: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_23),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_14),E))) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_23),Y_14)) ) ).

tff(fact_1630_minf_I8_J,axiom,
    ! [T_8: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Z))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,T_8),X_2)) ) ).

tff(fact_1631_minf_I8_J,axiom,
    ! [T_8: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_2),Z))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,T_8),X_2)) ) ).

tff(fact_1632_minf_I8_J,axiom,
    ! [T_8: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Z))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,T_8),X_2)) ) ).

tff(fact_1633_minf_I8_J,axiom,
    ! [T_8: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Z))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_8),X_2)) ) ).

tff(fact_1634_minf_I8_J,axiom,
    ! [T_8: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T_8),X_2)) ) ).

tff(fact_1635_pinf_I8_J,axiom,
    ! [T_7: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X_2))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,T_7),X_2)) ) ).

tff(fact_1636_pinf_I8_J,axiom,
    ! [T_7: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X_2))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,T_7),X_2)) ) ).

tff(fact_1637_pinf_I8_J,axiom,
    ! [T_7: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,T_7),X_2)) ) ).

tff(fact_1638_pinf_I8_J,axiom,
    ! [T_7: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X_2))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_7),X_2)) ) ).

tff(fact_1639_pinf_I8_J,axiom,
    ! [T_7: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T_7),X_2)) ) ).

tff(fact_1640_minf_I6_J,axiom,
    ! [T_6: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Z))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_2),T_6)) ) ).

tff(fact_1641_minf_I6_J,axiom,
    ! [T_6: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_2),Z))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_2),T_6)) ) ).

tff(fact_1642_minf_I6_J,axiom,
    ! [T_6: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),T_6)) ) ).

tff(fact_1643_minf_I6_J,axiom,
    ! [T_6: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Z))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),T_6)) ) ).

tff(fact_1644_minf_I6_J,axiom,
    ! [T_6: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),T_6)) ) ).

tff(fact_1645_pinf_I6_J,axiom,
    ! [T_5: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X_2))
     => ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_2),T_5)) ) ).

tff(fact_1646_pinf_I6_J,axiom,
    ! [T_5: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X_2))
     => ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_2),T_5)) ) ).

tff(fact_1647_pinf_I6_J,axiom,
    ! [T_5: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_2))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),T_5)) ) ).

tff(fact_1648_pinf_I6_J,axiom,
    ! [T_5: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X_2))
     => ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),T_5)) ) ).

tff(fact_1649_pinf_I6_J,axiom,
    ! [T_5: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),T_5)) ) ).

tff(fact_1650_minusinfinity,axiom,
    ! [P_1: fun_int_bool,P1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int,K: int] :
            ( hBOOL(hAPP_int_bool(P1,X_2))
          <=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z: int] :
            ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
              <=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
         => ( ? [X1: int] : hBOOL(hAPP_int_bool(P1,X1))
           => ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).

tff(fact_1651_plusinfinity,axiom,
    ! [P_1: fun_int_bool,P_2: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int,K: int] :
            ( hBOOL(hAPP_int_bool(P_2,X_2))
          <=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z: int] :
            ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
              <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
         => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_2,X1))
           => ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).

tff(fact_1652_tsub__def,axiom,
    ! [Y: int,X: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => ( hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => ( hAPP_int_int(nat_tsub(X),Y) = zero_zero_int ) ) ) ).

tff(fact_1653_zfact_Osimps,axiom,
    ! [N_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
       => ( hAPP_int_int(zfact,N_1) = one_one_int ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
       => ( hAPP_int_int(zfact,N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int))) ) ) ) ).

tff(fact_1654_floor__le__number__of,axiom,
    ! [X_1: rat,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim791455193or_rat(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V)),one_one_rat))) ) ).

tff(fact_1655_floor__le__number__of,axiom,
    ! [X_1: real,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V)),one_one_real))) ) ).

tff(fact_1656_floor__zero,axiom,
    archim791455193or_rat(zero_zero_rat) = zero_zero_int ).

tff(fact_1657_floor__zero,axiom,
    archim1246769320r_real(zero_zero_real) = zero_zero_int ).

tff(fact_1658_floor__less__cancel,axiom,
    ! [X_22: rat,Y_13: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim791455193or_rat(X_22)),archim791455193or_rat(Y_13)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_22),Y_13)) ) ).

tff(fact_1659_floor__less__cancel,axiom,
    ! [X_22: real,Y_13: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_22)),archim1246769320r_real(Y_13)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Y_13)) ) ).

tff(fact_1660_floor__mono,axiom,
    ! [X_21: rat,Y_12: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_21),Y_12))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim791455193or_rat(X_21)),archim791455193or_rat(Y_12))) ) ).

tff(fact_1661_floor__mono,axiom,
    ! [X_21: real,Y_12: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_21),Y_12))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_21)),archim1246769320r_real(Y_12))) ) ).

tff(fact_1662_floor__one,axiom,
    archim791455193or_rat(one_one_rat) = one_one_int ).

tff(fact_1663_floor__one,axiom,
    archim1246769320r_real(one_one_real) = one_one_int ).

tff(fact_1664_floor__number__of,axiom,
    ! [V_8: int] : archim791455193or_rat(hAPP_int_rat(number_number_of_rat,V_8)) = hAPP_int_int(number_number_of_int,V_8) ).

tff(fact_1665_floor__number__of,axiom,
    ! [V_8: int] : archim1246769320r_real(hAPP_int_real(number267125858f_real,V_8)) = hAPP_int_int(number_number_of_int,V_8) ).

tff(fact_1666_floor__less__zero,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim791455193or_rat(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),zero_zero_rat)) ) ).

tff(fact_1667_floor__less__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ).

tff(fact_1668_zero__le__floor,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),archim791455193or_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_1)) ) ).

tff(fact_1669_zero__le__floor,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) ) ).

tff(fact_1670_zero__less__floor,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),archim791455193or_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),X_1)) ) ).

tff(fact_1671_zero__less__floor,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ).

tff(fact_1672_floor__le__zero,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim791455193or_rat(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),one_one_rat)) ) ).

tff(fact_1673_floor__le__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ).

tff(fact_1674_floor__less__one,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim791455193or_rat(X_1)),one_one_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),one_one_rat)) ) ).

tff(fact_1675_floor__less__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),one_one_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ).

tff(fact_1676_one__le__floor,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),archim791455193or_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),X_1)) ) ).

tff(fact_1677_one__le__floor,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ).

tff(fact_1678_floor__less__number__of,axiom,
    ! [X_1: rat,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim791455193or_rat(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),hAPP_int_rat(number_number_of_rat,V))) ) ).

tff(fact_1679_floor__less__number__of,axiom,
    ! [X_1: real,V: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),hAPP_int_int(number_number_of_int,V)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,V))) ) ).

tff(fact_1680_number__of__le__floor,axiom,
    ! [V: int,X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),archim791455193or_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,V)),X_1)) ) ).

tff(fact_1681_number__of__le__floor,axiom,
    ! [V: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,V)),X_1)) ) ).

tff(fact_1682_floor__add__one,axiom,
    ! [X_20: rat] : archim791455193or_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_20),one_one_rat)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim791455193or_rat(X_20)),one_one_int) ).

tff(fact_1683_floor__add__one,axiom,
    ! [X_20: real] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_20),one_one_real)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X_20)),one_one_int) ).

tff(fact_1684_floor__add__number__of,axiom,
    ! [X_19: rat,V_7: int] : archim791455193or_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_19),hAPP_int_rat(number_number_of_rat,V_7))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim791455193or_rat(X_19)),hAPP_int_int(number_number_of_int,V_7)) ).

tff(fact_1685_floor__add__number__of,axiom,
    ! [X_19: real,V_7: int] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_19),hAPP_int_real(number267125858f_real,V_7))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X_19)),hAPP_int_int(number_number_of_int,V_7)) ).

tff(fact_1686_floor__diff__one,axiom,
    ! [X_18: rat] : archim791455193or_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_18),one_one_rat)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim791455193or_rat(X_18)),one_one_int) ).

tff(fact_1687_floor__diff__one,axiom,
    ! [X_18: real] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_18),one_one_real)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X_18)),one_one_int) ).

tff(fact_1688_floor__diff__number__of,axiom,
    ! [X_17: rat,V_6: int] : archim791455193or_rat(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_17),hAPP_int_rat(number_number_of_rat,V_6))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim791455193or_rat(X_17)),hAPP_int_int(number_number_of_int,V_6)) ).

tff(fact_1689_floor__diff__number__of,axiom,
    ! [X_17: real,V_6: int] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_17),hAPP_int_real(number267125858f_real,V_6))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X_17)),hAPP_int_int(number_number_of_int,V_6)) ).

tff(fact_1690_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(nat_tsub(X),Y))) ) ) ).

tff(fact_1691_tsub__eq,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
     => ( hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) ) ).

tff(fact_1692_number__of__less__floor,axiom,
    ! [V: int,X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,V)),archim791455193or_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V)),one_one_rat)),X_1)) ) ).

tff(fact_1693_number__of__less__floor,axiom,
    ! [V: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,V)),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V)),one_one_real)),X_1)) ) ).

tff(fact_1694_le__mult__floor,axiom,
    ! [B_1: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,archim1246769320r_real(A_2)),archim1246769320r_real(B_1))),archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),B_1)))) ) ) ).

tff(fact_1695_posDivAlg__correct,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),posDivAlg(A_2,B_1))) ) ) ).

tff(fact_1696_incr__lemma,axiom,
    ! [Z_1: int,X: int,D_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3)))) ) ).

tff(fact_1697_decr__lemma,axiom,
    ! [X: int,Z_1: int,D_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3))),Z_1)) ) ).

tff(fact_1698_Ints__odd__less__0,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),Aa)),Aa)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),zero_zero_real)) ) ) ).

tff(fact_1699_Ints__odd__less__0,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Aa)),Aa)),zero_zero_int))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ) ).

tff(fact_1700_Ints__odd__less__0,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),Aa)),Aa)),zero_zero_rat))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),zero_zero_rat)) ) ) ).

tff(fact_1701_zminus1__lemma,axiom,
    ! [Aa: int,Ba: int,Qa: int,R_2: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(Aa,Ba),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Qa),R_2)))
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,Aa),Ba),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Qa)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Qa)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),R_2))))) ) ).

tff(fact_1702_d22set__mem,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ba))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(d22set,Aa))) ) ) ).

tff(fact_1703_zero__le__ceiling,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),archim1419379755ng_rat(X_1)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,min)),X_1)) ) ).

tff(fact_1704_zero__le__ceiling,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1)) ) ).

tff(fact_1705_ceiling__less__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(number267125858f_real,min))) ) ).

tff(fact_1706_ceiling__less__zero,axiom,
    ! [X_1: rat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1419379755ng_rat(X_1)),zero_zero_int))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),hAPP_int_rat(number_number_of_rat,min))) ) ).

tff(fact_1707_split__zmod,axiom,
    ! [P_1: fun_int_bool,N: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(N),K_1)))
    <=> ( ( ( K_1 = zero_zero_int )
         => hBOOL(hAPP_int_bool(P_1,N)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
                & ( N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,J_1)) ) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
                & ( N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,J_1)) ) ) ) ) ).

tff(fact_1708_real__mult__le__cancel__iff1,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ).

tff(fact_1709_real__mult__le__cancel__iff2,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ).

tff(fact_1710_real__le__eq__diff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_2)),zero_zero_real)) ) ).

tff(fact_1711_zmod__zminus2__not__zero,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) != zero_zero_int )
     => ( hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int ) ) ).

tff(fact_1712_zmod__zminus1__not__zero,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,K_2)),L_2) != zero_zero_int )
     => ( hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int ) ) ).

tff(fact_1713_real__mult__left__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( C != zero_zero_real )
     => ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Aa) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),Ba) )
      <=> ( Aa = Ba ) ) ) ).

tff(fact_1714_real__mult__right__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( C != zero_zero_real )
     => ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),C) )
      <=> ( Aa = Ba ) ) ) ).

tff(fact_1715_real__two__squares__add__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_1716_real__mult__less__mono2,axiom,
    ! [X: real,Y: real,Z_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),Y))) ) ) ).

tff(fact_1717_real__mult__order,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))) ) ) ).

tff(fact_1718_real__mult__less__iff1,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ).

tff(fact_1719_real__zero__not__eq__one,axiom,
    zero_zero_real != one_one_real ).

tff(fact_1720_mod__minus__cong,axiom,
    ! [A_114: int,B_85: int,A_113: int] :
      ( ( hAPP_int_int(div_mod_int(A_114),B_85) = hAPP_int_int(div_mod_int(A_113),B_85) )
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_114)),B_85) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_113)),B_85) ) ) ).

tff(fact_1721_mod__mod__trivial,axiom,
    ! [A_112: quickcheck_code_int,B_84: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_112),B_84)),B_84) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_112),B_84) ).

tff(fact_1722_mod__mod__trivial,axiom,
    ! [A_112: code_code_numeral,B_84: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_112),B_84)),B_84) = hAPP_C498520661umeral(div_mo1740067990umeral(A_112),B_84) ).

tff(fact_1723_mod__mod__trivial,axiom,
    ! [A_112: nat,B_84: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_112),B_84)),B_84) = hAPP_nat_nat(div_mod_nat(A_112),B_84) ).

tff(fact_1724_mod__mod__trivial,axiom,
    ! [A_112: int,B_84: int] : hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_112),B_84)),B_84) = hAPP_int_int(div_mod_int(A_112),B_84) ).

tff(fact_1725_zminus__zmod,axiom,
    ! [X: int,M_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X),M_1))),M_1) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X)),M_1) ).

tff(fact_1726_zmod__zminus__zminus,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_2),B_1)) ).

tff(fact_1727_mod__minus__eq,axiom,
    ! [A_111: int,B_83: int] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_111)),B_83) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_111),B_83))),B_83) ).

tff(fact_1728_zmod__zminus2,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(div_mod_int(A_2),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_2)),B_1)) ).

tff(fact_1729_arith__simps_I30_J,axiom,
    ! [W_7: int] : hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(number_number_of_rat,W_7)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(uminus_uminus_int,W_7)) ).

tff(fact_1730_arith__simps_I30_J,axiom,
    ! [W_7: int] : hAPP_complex_complex(uminus473333897omplex,hAPP_int_complex(number528085621omplex,W_7)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(uminus_uminus_int,W_7)) ).

tff(fact_1731_arith__simps_I30_J,axiom,
    ! [W_7: int] : hAPP_real_real(uminus_uminus_real,hAPP_int_real(number267125858f_real,W_7)) = hAPP_int_real(number267125858f_real,hAPP_int_int(uminus_uminus_int,W_7)) ).

tff(fact_1732_arith__simps_I30_J,axiom,
    ! [W_7: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W_7)) = hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W_7)) ).

tff(fact_1733_number__of__minus,axiom,
    ! [W_6: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(uminus_uminus_int,W_6)) = hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(number_number_of_rat,W_6)) ).

tff(fact_1734_number__of__minus,axiom,
    ! [W_6: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(uminus_uminus_int,W_6)) = hAPP_complex_complex(uminus473333897omplex,hAPP_int_complex(number528085621omplex,W_6)) ).

tff(fact_1735_number__of__minus,axiom,
    ! [W_6: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(uminus_uminus_int,W_6)) = hAPP_real_real(uminus_uminus_real,hAPP_int_real(number267125858f_real,W_6)) ).

tff(fact_1736_number__of__minus,axiom,
    ! [W_6: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W_6)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W_6)) ).

tff(fact_1737_Int_OMin__def,axiom,
    min = hAPP_int_int(uminus_uminus_int,one_one_int) ).

tff(fact_1738_mult__Min,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,min),K_2) = hAPP_int_int(uminus_uminus_int,K_2) ).

tff(fact_1739_real__le__refl,axiom,
    ! [W: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).

tff(fact_1740_real__le__linear,axiom,
    ! [Z_1: real,W: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W))
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) ) ).

tff(fact_1741_real__add__left__mono,axiom,
    ! [Z_1: real,X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),Y))) ) ).

tff(fact_1742_real__le__trans,axiom,
    ! [K_2: real,I: real,J_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K_2)) ) ) ).

tff(fact_1743_real__le__antisym,axiom,
    ! [Z_1: real,W: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1))
       => ( Z_1 = W ) ) ) ).

tff(fact_1744_abs__ge__minus__self,axiom,
    ! [A_110: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,A_110)),hAPP_rat_rat(abs_abs_rat,A_110))) ).

tff(fact_1745_abs__ge__minus__self,axiom,
    ! [A_110: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,A_110)),hAPP_real_real(abs_abs_real,A_110))) ).

tff(fact_1746_abs__ge__minus__self,axiom,
    ! [A_110: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,A_110)),hAPP_int_int(abs_abs_int,A_110))) ).

tff(fact_1747_abs__le__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,Aa)),Ba))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba))
        & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),Ba)) ) ) ).

tff(fact_1748_abs__le__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Aa)),Ba))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
        & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Aa)),Ba)) ) ) ).

tff(fact_1749_abs__le__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,Aa)),Ba))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,Aa)),Ba)) ) ) ).

tff(fact_1750_abs__leI,axiom,
    ! [A_109: rat,B_82: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_109),B_82))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,A_109)),B_82))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_109)),B_82)) ) ) ).

tff(fact_1751_abs__leI,axiom,
    ! [A_109: real,B_82: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_109),B_82))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,A_109)),B_82))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_109)),B_82)) ) ) ).

tff(fact_1752_abs__leI,axiom,
    ! [A_109: int,B_82: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_109),B_82))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,A_109)),B_82))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_109)),B_82)) ) ) ).

tff(fact_1753_abs__le__D2,axiom,
    ! [A_108: rat,B_81: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_108)),B_81))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,A_108)),B_81)) ) ).

tff(fact_1754_abs__le__D2,axiom,
    ! [A_108: real,B_81: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_108)),B_81))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,A_108)),B_81)) ) ).

tff(fact_1755_abs__le__D2,axiom,
    ! [A_108: int,B_81: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_108)),B_81))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,A_108)),B_81)) ) ).

tff(fact_1756_Ints__minus,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_real_real(uminus_uminus_real,Aa)),ring_1_Ints_real)) ) ).

tff(fact_1757_Ints__minus,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(uminus_uminus_int,Aa)),ring_1_Ints_int)) ) ).

tff(fact_1758_Ints__minus,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),ring_1_Ints_rat)) ) ).

tff(fact_1759_Ints__minus,axiom,
    ! [Aa: complex] :
      ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Aa),ring_1_Ints_complex))
     => hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,hAPP_complex_complex(uminus473333897omplex,Aa)),ring_1_Ints_complex)) ) ).

tff(fact_1760_real__add__mult__distrib,axiom,
    ! [Z1: real,Z2: real,W: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z1),Z2)),W) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),W)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),W)) ).

tff(fact_1761_real__mult__assoc,axiom,
    ! [Z1: real,Z2: real,Z3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),Z2)),Z3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),Z3)) ).

tff(fact_1762_neg__equal__iff__equal,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( hAPP_rat_rat(uminus_uminus_rat,Aa) = hAPP_rat_rat(uminus_uminus_rat,Ba) )
    <=> ( Aa = Ba ) ) ).

tff(fact_1763_neg__equal__iff__equal,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( hAPP_complex_complex(uminus473333897omplex,Aa) = hAPP_complex_complex(uminus473333897omplex,Ba) )
    <=> ( Aa = Ba ) ) ).

tff(fact_1764_neg__equal__iff__equal,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_real(uminus_uminus_real,Aa) = hAPP_real_real(uminus_uminus_real,Ba) )
    <=> ( Aa = Ba ) ) ).

tff(fact_1765_neg__equal__iff__equal,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(uminus_uminus_int,Aa) = hAPP_int_int(uminus_uminus_int,Ba) )
    <=> ( Aa = Ba ) ) ).

tff(fact_1766_minus__equation__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( hAPP_rat_rat(uminus_uminus_rat,Aa) = Ba )
    <=> ( hAPP_rat_rat(uminus_uminus_rat,Ba) = Aa ) ) ).

tff(fact_1767_minus__equation__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( hAPP_complex_complex(uminus473333897omplex,Aa) = Ba )
    <=> ( hAPP_complex_complex(uminus473333897omplex,Ba) = Aa ) ) ).

tff(fact_1768_minus__equation__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_real(uminus_uminus_real,Aa) = Ba )
    <=> ( hAPP_real_real(uminus_uminus_real,Ba) = Aa ) ) ).

tff(fact_1769_minus__equation__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(uminus_uminus_int,Aa) = Ba )
    <=> ( hAPP_int_int(uminus_uminus_int,Ba) = Aa ) ) ).

tff(fact_1770_equation__minus__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = hAPP_rat_rat(uminus_uminus_rat,Ba) )
    <=> ( Ba = hAPP_rat_rat(uminus_uminus_rat,Aa) ) ) ).

tff(fact_1771_equation__minus__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( Aa = hAPP_complex_complex(uminus473333897omplex,Ba) )
    <=> ( Ba = hAPP_complex_complex(uminus473333897omplex,Aa) ) ) ).

tff(fact_1772_equation__minus__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = hAPP_real_real(uminus_uminus_real,Ba) )
    <=> ( Ba = hAPP_real_real(uminus_uminus_real,Aa) ) ) ).

tff(fact_1773_equation__minus__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = hAPP_int_int(uminus_uminus_int,Ba) )
    <=> ( Ba = hAPP_int_int(uminus_uminus_int,Aa) ) ) ).

tff(fact_1774_real__mult__commute,axiom,
    ! [Z_1: real,W: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1) ).

tff(fact_1775_abs__minus__cancel,axiom,
    ! [A_107: rat] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(uminus_uminus_rat,A_107)) = hAPP_rat_rat(abs_abs_rat,A_107) ).

tff(fact_1776_abs__minus__cancel,axiom,
    ! [A_107: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(uminus_uminus_real,A_107)) = hAPP_real_real(abs_abs_real,A_107) ).

tff(fact_1777_abs__minus__cancel,axiom,
    ! [A_107: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(uminus_uminus_int,A_107)) = hAPP_int_int(abs_abs_int,A_107) ).

tff(fact_1778_abs__idempotent,axiom,
    ! [A_106: rat] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(abs_abs_rat,A_106)) = hAPP_rat_rat(abs_abs_rat,A_106) ).

tff(fact_1779_abs__idempotent,axiom,
    ! [A_106: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(abs_abs_real,A_106)) = hAPP_real_real(abs_abs_real,A_106) ).

tff(fact_1780_abs__idempotent,axiom,
    ! [A_106: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(abs_abs_int,A_106)) = hAPP_int_int(abs_abs_int,A_106) ).

tff(fact_1781_zminus__zminus,axiom,
    ! [Z_1: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)) = Z_1 ).

tff(fact_1782_minus__minus,axiom,
    ! [A_105: rat] : hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(uminus_uminus_rat,A_105)) = A_105 ).

tff(fact_1783_minus__minus,axiom,
    ! [A_105: complex] : hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(uminus473333897omplex,A_105)) = A_105 ).

tff(fact_1784_minus__minus,axiom,
    ! [A_105: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(uminus_uminus_real,A_105)) = A_105 ).

tff(fact_1785_minus__minus,axiom,
    ! [A_105: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,A_105)) = A_105 ).

tff(fact_1786_real__mult__1,axiom,
    ! [Z_1: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1) = Z_1 ).

tff(fact_1787_abs__less__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,Aa)),Ba))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba))
        & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),Ba)) ) ) ).

tff(fact_1788_abs__less__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Aa)),Ba))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
        & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Aa)),Ba)) ) ) ).

tff(fact_1789_abs__less__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Aa)),Ba))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,Aa)),Ba)) ) ) ).

tff(fact_1790_less__eq__real__def,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_1791_real__less__def,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_1792_abs__minus__le__zero,axiom,
    ! [A_104: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(abs_abs_rat,A_104))),zero_zero_rat)) ).

tff(fact_1793_abs__minus__le__zero,axiom,
    ! [A_104: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(abs_abs_real,A_104))),zero_zero_real)) ).

tff(fact_1794_abs__minus__le__zero,axiom,
    ! [A_104: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(abs_abs_int,A_104))),zero_zero_int)) ).

tff(fact_1795_abs__of__nonpos,axiom,
    ! [A_103: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_103),zero_zero_rat))
     => ( hAPP_rat_rat(abs_abs_rat,A_103) = hAPP_rat_rat(uminus_uminus_rat,A_103) ) ) ).

tff(fact_1796_abs__of__nonpos,axiom,
    ! [A_103: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_103),zero_zero_real))
     => ( hAPP_real_real(abs_abs_real,A_103) = hAPP_real_real(uminus_uminus_real,A_103) ) ) ).

tff(fact_1797_abs__of__nonpos,axiom,
    ! [A_103: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_103),zero_zero_int))
     => ( hAPP_int_int(abs_abs_int,A_103) = hAPP_int_int(uminus_uminus_int,A_103) ) ) ).

tff(fact_1798_abs__of__neg,axiom,
    ! [A_102: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_102),zero_zero_rat))
     => ( hAPP_rat_rat(abs_abs_rat,A_102) = hAPP_rat_rat(uminus_uminus_rat,A_102) ) ) ).

tff(fact_1799_abs__of__neg,axiom,
    ! [A_102: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_102),zero_zero_real))
     => ( hAPP_real_real(abs_abs_real,A_102) = hAPP_real_real(uminus_uminus_real,A_102) ) ) ).

tff(fact_1800_abs__of__neg,axiom,
    ! [A_102: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_102),zero_zero_int))
     => ( hAPP_int_int(abs_abs_int,A_102) = hAPP_int_int(uminus_uminus_int,A_102) ) ) ).

tff(fact_1801_abs__if,axiom,
    ! [A_101: rat] :
      ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_101),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,A_101) = hAPP_rat_rat(uminus_uminus_rat,A_101) ) )
      & ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_101),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,A_101) = A_101 ) ) ) ).

tff(fact_1802_abs__if,axiom,
    ! [A_101: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_101),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,A_101) = hAPP_real_real(uminus_uminus_real,A_101) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_101),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,A_101) = A_101 ) ) ) ).

tff(fact_1803_abs__if,axiom,
    ! [A_101: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_101),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,A_101) = hAPP_int_int(uminus_uminus_int,A_101) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_101),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,A_101) = A_101 ) ) ) ).

tff(fact_1804_abs__minus__one,axiom,
    hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(number_number_of_rat,min)) = one_one_rat ).

tff(fact_1805_abs__minus__one,axiom,
    hAPP_real_real(abs_abs_real,hAPP_int_real(number267125858f_real,min)) = one_one_real ).

tff(fact_1806_abs__minus__one,axiom,
    hAPP_int_int(abs_abs_int,hAPP_int_int(number_number_of_int,min)) = one_one_int ).

tff(fact_1807_minus__number__of__mult,axiom,
    ! [W_5: int,Z_9: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(number_number_of_rat,W_5))),Z_9) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(uminus_uminus_int,W_5))),Z_9) ).

tff(fact_1808_minus__number__of__mult,axiom,
    ! [W_5: int,Z_9: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,hAPP_int_complex(number528085621omplex,W_5))),Z_9) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(uminus_uminus_int,W_5))),Z_9) ).

tff(fact_1809_minus__number__of__mult,axiom,
    ! [W_5: int,Z_9: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_int_real(number267125858f_real,W_5))),Z_9) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(uminus_uminus_int,W_5))),Z_9) ).

tff(fact_1810_minus__number__of__mult,axiom,
    ! [W_5: int,Z_9: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W_5))),Z_9) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W_5))),Z_9) ).

tff(fact_1811_mult__minus1,axiom,
    ! [Z_8: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,min)),Z_8) = hAPP_rat_rat(uminus_uminus_rat,Z_8) ).

tff(fact_1812_mult__minus1,axiom,
    ! [Z_8: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,min)),Z_8) = hAPP_complex_complex(uminus473333897omplex,Z_8) ).

tff(fact_1813_mult__minus1,axiom,
    ! [Z_8: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,min)),Z_8) = hAPP_real_real(uminus_uminus_real,Z_8) ).

tff(fact_1814_mult__minus1,axiom,
    ! [Z_8: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,min)),Z_8) = hAPP_int_int(uminus_uminus_int,Z_8) ).

tff(fact_1815_mult__minus1__right,axiom,
    ! [Z_7: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_7),hAPP_int_rat(number_number_of_rat,min)) = hAPP_rat_rat(uminus_uminus_rat,Z_7) ).

tff(fact_1816_mult__minus1__right,axiom,
    ! [Z_7: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_7),hAPP_int_complex(number528085621omplex,min)) = hAPP_complex_complex(uminus473333897omplex,Z_7) ).

tff(fact_1817_mult__minus1__right,axiom,
    ! [Z_7: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_7),hAPP_int_real(number267125858f_real,min)) = hAPP_real_real(uminus_uminus_real,Z_7) ).

tff(fact_1818_mult__minus1__right,axiom,
    ! [Z_7: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_7),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(uminus_uminus_int,Z_7) ).

tff(fact_1819_number__of__Min,axiom,
    hAPP_int_rat(number_number_of_rat,min) = hAPP_rat_rat(uminus_uminus_rat,one_one_rat) ).

tff(fact_1820_number__of__Min,axiom,
    hAPP_int_complex(number528085621omplex,min) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).

tff(fact_1821_number__of__Min,axiom,
    hAPP_int_real(number267125858f_real,min) = hAPP_real_real(uminus_uminus_real,one_one_real) ).

tff(fact_1822_number__of__Min,axiom,
    hAPP_int_int(number_number_of_int,min) = hAPP_int_int(uminus_uminus_int,one_one_int) ).

tff(fact_1823_arith__simps_I31_J,axiom,
    hAPP_rat_rat(uminus_uminus_rat,one_one_rat) = hAPP_int_rat(number_number_of_rat,min) ).

tff(fact_1824_arith__simps_I31_J,axiom,
    hAPP_complex_complex(uminus473333897omplex,one_one_complex) = hAPP_int_complex(number528085621omplex,min) ).

tff(fact_1825_arith__simps_I31_J,axiom,
    hAPP_real_real(uminus_uminus_real,one_one_real) = hAPP_int_real(number267125858f_real,min) ).

tff(fact_1826_arith__simps_I31_J,axiom,
    hAPP_int_int(uminus_uminus_int,one_one_int) = hAPP_int_int(number_number_of_int,min) ).

tff(fact_1827_zabs__def,axiom,
    ! [I: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,I) = hAPP_int_int(uminus_uminus_int,I) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,I) = I ) ) ) ).

tff(fact_1828_zmod__minus1__right,axiom,
    ! [A_2: int] : hAPP_int_int(div_mod_int(A_2),hAPP_int_int(number_number_of_int,min)) = zero_zero_int ).

tff(fact_1829_zmod__zminus1__eq__if,axiom,
    ! [A_2: int,B_1: int] :
      ( ( ( hAPP_int_int(div_mod_int(A_2),B_1) = zero_zero_int )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_2)),B_1) = zero_zero_int ) )
      & ( ( hAPP_int_int(div_mod_int(A_2),B_1) != zero_zero_int )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_2)),B_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),hAPP_int_int(div_mod_int(A_2),B_1)) ) ) ) ).

tff(fact_1830_zmod__zminus2__eq__if,axiom,
    ! [A_2: int,B_1: int] :
      ( ( ( hAPP_int_int(div_mod_int(A_2),B_1) = zero_zero_int )
       => ( hAPP_int_int(div_mod_int(A_2),hAPP_int_int(uminus_uminus_int,B_1)) = zero_zero_int ) )
      & ( ( hAPP_int_int(div_mod_int(A_2),B_1) != zero_zero_int )
       => ( hAPP_int_int(div_mod_int(A_2),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_2),B_1)),B_1) ) ) ) ).

tff(fact_1831_abs__eq__1__iff,axiom,
    ! [Z_2: int] :
      ( ( hAPP_int_int(abs_abs_int,Z_2) = one_one_int )
    <=> ( ( Z_2 = one_one_int )
        | ( Z_2 = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_1832_mod__0,axiom,
    ! [A_100: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(zero_z891286103de_int),A_100) = zero_z891286103de_int ).

tff(fact_1833_mod__0,axiom,
    ! [A_100: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(zero_z126310315umeral),A_100) = zero_z126310315umeral ).

tff(fact_1834_mod__0,axiom,
    ! [A_100: nat] : hAPP_nat_nat(div_mod_nat(zero_zero_nat),A_100) = zero_zero_nat ).

tff(fact_1835_mod__0,axiom,
    ! [A_100: int] : hAPP_int_int(div_mod_int(zero_zero_int),A_100) = zero_zero_int ).

tff(fact_1836_mod__by__0,axiom,
    ! [A_99: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_99),zero_z891286103de_int) = A_99 ).

tff(fact_1837_mod__by__0,axiom,
    ! [A_99: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(A_99),zero_z126310315umeral) = A_99 ).

tff(fact_1838_mod__by__0,axiom,
    ! [A_99: nat] : hAPP_nat_nat(div_mod_nat(A_99),zero_zero_nat) = A_99 ).

tff(fact_1839_mod__by__0,axiom,
    ! [A_99: int] : hAPP_int_int(div_mod_int(A_99),zero_zero_int) = A_99 ).

tff(fact_1840_mod__self,axiom,
    ! [A_98: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_98),A_98) = zero_z891286103de_int ).

tff(fact_1841_mod__self,axiom,
    ! [A_98: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(A_98),A_98) = zero_z126310315umeral ).

tff(fact_1842_mod__self,axiom,
    ! [A_98: nat] : hAPP_nat_nat(div_mod_nat(A_98),A_98) = zero_zero_nat ).

tff(fact_1843_mod__self,axiom,
    ! [A_98: int] : hAPP_int_int(div_mod_int(A_98),A_98) = zero_zero_int ).

tff(fact_1844_abs__eq__0,axiom,
    ! [Aa: rat] :
      ( ( hAPP_rat_rat(abs_abs_rat,Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1845_abs__eq__0,axiom,
    ! [Aa: real] :
      ( ( hAPP_real_real(abs_abs_real,Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1846_abs__eq__0,axiom,
    ! [Aa: int] :
      ( ( hAPP_int_int(abs_abs_int,Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1847_abs__zero,axiom,
    hAPP_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

tff(fact_1848_abs__zero,axiom,
    hAPP_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

tff(fact_1849_abs__zero,axiom,
    hAPP_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

tff(fact_1850_abs__le__D1,axiom,
    ! [A_97: rat,B_80: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_97)),B_80))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_97),B_80)) ) ).

tff(fact_1851_abs__le__D1,axiom,
    ! [A_97: real,B_80: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_97)),B_80))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_97),B_80)) ) ).

tff(fact_1852_abs__le__D1,axiom,
    ! [A_97: int,B_80: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_97)),B_80))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_97),B_80)) ) ).

tff(fact_1853_abs__ge__self,axiom,
    ! [A_96: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_96),hAPP_rat_rat(abs_abs_rat,A_96))) ).

tff(fact_1854_abs__ge__self,axiom,
    ! [A_96: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_96),hAPP_real_real(abs_abs_real,A_96))) ).

tff(fact_1855_abs__ge__self,axiom,
    ! [A_96: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_96),hAPP_int_int(abs_abs_int,A_96))) ).

tff(fact_1856_mod__mult__cong,axiom,
    ! [B_79: quickcheck_code_int,B_78: quickcheck_code_int,A_95: quickcheck_code_int,C_35: quickcheck_code_int,A_94: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(A_95),C_35) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_94),C_35) )
     => ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_79),C_35) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_78),C_35) )
       => ( hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_95),B_79)),C_35) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_94),B_78)),C_35) ) ) ) ).

tff(fact_1857_mod__mult__cong,axiom,
    ! [B_79: code_code_numeral,B_78: code_code_numeral,A_95: code_code_numeral,C_35: code_code_numeral,A_94: code_code_numeral] :
      ( ( hAPP_C498520661umeral(div_mo1740067990umeral(A_95),C_35) = hAPP_C498520661umeral(div_mo1740067990umeral(A_94),C_35) )
     => ( ( hAPP_C498520661umeral(div_mo1740067990umeral(B_79),C_35) = hAPP_C498520661umeral(div_mo1740067990umeral(B_78),C_35) )
       => ( hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_95),B_79)),C_35) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_94),B_78)),C_35) ) ) ) ).

tff(fact_1858_mod__mult__cong,axiom,
    ! [B_79: nat,B_78: nat,A_95: nat,C_35: nat,A_94: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(A_95),C_35) = hAPP_nat_nat(div_mod_nat(A_94),C_35) )
     => ( ( hAPP_nat_nat(div_mod_nat(B_79),C_35) = hAPP_nat_nat(div_mod_nat(B_78),C_35) )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_95),B_79)),C_35) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_94),B_78)),C_35) ) ) ) ).

tff(fact_1859_mod__mult__cong,axiom,
    ! [B_79: int,B_78: int,A_95: int,C_35: int,A_94: int] :
      ( ( hAPP_int_int(div_mod_int(A_95),C_35) = hAPP_int_int(div_mod_int(A_94),C_35) )
     => ( ( hAPP_int_int(div_mod_int(B_79),C_35) = hAPP_int_int(div_mod_int(B_78),C_35) )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_95),B_79)),C_35) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_94),B_78)),C_35) ) ) ) ).

tff(fact_1860_zmod__simps_I4_J,axiom,
    ! [A_93: quickcheck_code_int,C_34: quickcheck_code_int,B_77: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_93),C_34)),B_77)),C_34) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_93),B_77)),C_34) ).

tff(fact_1861_zmod__simps_I4_J,axiom,
    ! [A_93: code_code_numeral,C_34: code_code_numeral,B_77: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_93),C_34)),B_77)),C_34) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_93),B_77)),C_34) ).

tff(fact_1862_zmod__simps_I4_J,axiom,
    ! [A_93: nat,C_34: nat,B_77: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_93),C_34)),B_77)),C_34) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_93),B_77)),C_34) ).

tff(fact_1863_zmod__simps_I4_J,axiom,
    ! [A_93: int,C_34: int,B_77: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_93),C_34)),B_77)),C_34) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_93),B_77)),C_34) ).

tff(fact_1864_mod__mult__mult2,axiom,
    ! [A_92: quickcheck_code_int,C_33: quickcheck_code_int,B_76: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_92),C_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_76),C_33)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_92),B_76)),C_33) ).

tff(fact_1865_mod__mult__mult2,axiom,
    ! [A_92: code_code_numeral,C_33: code_code_numeral,B_76: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_92),C_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_76),C_33)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_92),B_76)),C_33) ).

tff(fact_1866_mod__mult__mult2,axiom,
    ! [A_92: nat,C_33: nat,B_76: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_92),C_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_76),C_33)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_92),B_76)),C_33) ).

tff(fact_1867_mod__mult__mult2,axiom,
    ! [A_92: int,C_33: int,B_76: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_92),C_33)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_76),C_33)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_92),B_76)),C_33) ).

tff(fact_1868_mod__mult__mult1,axiom,
    ! [C_32: quickcheck_code_int,A_91: quickcheck_code_int,B_75: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_32),A_91)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_32),B_75)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_32),hAPP_Q1762011733de_int(div_mo231679042de_int(A_91),B_75)) ).

tff(fact_1869_mod__mult__mult1,axiom,
    ! [C_32: code_code_numeral,A_91: code_code_numeral,B_75: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_32),A_91)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_32),B_75)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_32),hAPP_C498520661umeral(div_mo1740067990umeral(A_91),B_75)) ).

tff(fact_1870_mod__mult__mult1,axiom,
    ! [C_32: nat,A_91: nat,B_75: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_32),A_91)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_32),B_75)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_32),hAPP_nat_nat(div_mod_nat(A_91),B_75)) ).

tff(fact_1871_mod__mult__mult1,axiom,
    ! [C_32: int,A_91: int,B_75: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_32),A_91)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_32),B_75)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_32),hAPP_int_int(div_mod_int(A_91),B_75)) ).

tff(fact_1872_mod__mult__eq,axiom,
    ! [A_90: quickcheck_code_int,B_74: quickcheck_code_int,C_31: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_90),B_74)),C_31) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_90),C_31)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_74),C_31))),C_31) ).

tff(fact_1873_mod__mult__eq,axiom,
    ! [A_90: code_code_numeral,B_74: code_code_numeral,C_31: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_90),B_74)),C_31) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_90),C_31)),hAPP_C498520661umeral(div_mo1740067990umeral(B_74),C_31))),C_31) ).

tff(fact_1874_mod__mult__eq,axiom,
    ! [A_90: nat,B_74: nat,C_31: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_90),B_74)),C_31) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_90),C_31)),hAPP_nat_nat(div_mod_nat(B_74),C_31))),C_31) ).

tff(fact_1875_mod__mult__eq,axiom,
    ! [A_90: int,B_74: int,C_31: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_90),B_74)),C_31) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_90),C_31)),hAPP_int_int(div_mod_int(B_74),C_31))),C_31) ).

tff(fact_1876_mod__mult__left__eq,axiom,
    ! [A_89: quickcheck_code_int,B_73: quickcheck_code_int,C_30: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_89),B_73)),C_30) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_89),C_30)),B_73)),C_30) ).

tff(fact_1877_mod__mult__left__eq,axiom,
    ! [A_89: code_code_numeral,B_73: code_code_numeral,C_30: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_89),B_73)),C_30) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_89),C_30)),B_73)),C_30) ).

tff(fact_1878_mod__mult__left__eq,axiom,
    ! [A_89: nat,B_73: nat,C_30: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_89),B_73)),C_30) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_89),C_30)),B_73)),C_30) ).

tff(fact_1879_mod__mult__left__eq,axiom,
    ! [A_89: int,B_73: int,C_30: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_89),B_73)),C_30) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_89),C_30)),B_73)),C_30) ).

tff(fact_1880_mod__mult__right__eq,axiom,
    ! [A_88: quickcheck_code_int,B_72: quickcheck_code_int,C_29: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_88),B_72)),C_29) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_88),hAPP_Q1762011733de_int(div_mo231679042de_int(B_72),C_29))),C_29) ).

tff(fact_1881_mod__mult__right__eq,axiom,
    ! [A_88: code_code_numeral,B_72: code_code_numeral,C_29: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_88),B_72)),C_29) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_88),hAPP_C498520661umeral(div_mo1740067990umeral(B_72),C_29))),C_29) ).

tff(fact_1882_mod__mult__right__eq,axiom,
    ! [A_88: nat,B_72: nat,C_29: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_88),B_72)),C_29) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_88),hAPP_nat_nat(div_mod_nat(B_72),C_29))),C_29) ).

tff(fact_1883_mod__mult__right__eq,axiom,
    ! [A_88: int,B_72: int,C_29: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_88),B_72)),C_29) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_88),hAPP_int_int(div_mod_int(B_72),C_29))),C_29) ).

tff(fact_1884_mod__add__cong,axiom,
    ! [B_71: quickcheck_code_int,B_70: quickcheck_code_int,A_87: quickcheck_code_int,C_28: quickcheck_code_int,A_86: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(A_87),C_28) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_86),C_28) )
     => ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_71),C_28) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_70),C_28) )
       => ( hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_87),B_71)),C_28) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_86),B_70)),C_28) ) ) ) ).

tff(fact_1885_mod__add__cong,axiom,
    ! [B_71: code_code_numeral,B_70: code_code_numeral,A_87: code_code_numeral,C_28: code_code_numeral,A_86: code_code_numeral] :
      ( ( hAPP_C498520661umeral(div_mo1740067990umeral(A_87),C_28) = hAPP_C498520661umeral(div_mo1740067990umeral(A_86),C_28) )
     => ( ( hAPP_C498520661umeral(div_mo1740067990umeral(B_71),C_28) = hAPP_C498520661umeral(div_mo1740067990umeral(B_70),C_28) )
       => ( hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_87),B_71)),C_28) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_86),B_70)),C_28) ) ) ) ).

tff(fact_1886_mod__add__cong,axiom,
    ! [B_71: nat,B_70: nat,A_87: nat,C_28: nat,A_86: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(A_87),C_28) = hAPP_nat_nat(div_mod_nat(A_86),C_28) )
     => ( ( hAPP_nat_nat(div_mod_nat(B_71),C_28) = hAPP_nat_nat(div_mod_nat(B_70),C_28) )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_87),B_71)),C_28) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_86),B_70)),C_28) ) ) ) ).

tff(fact_1887_mod__add__cong,axiom,
    ! [B_71: int,B_70: int,A_87: int,C_28: int,A_86: int] :
      ( ( hAPP_int_int(div_mod_int(A_87),C_28) = hAPP_int_int(div_mod_int(A_86),C_28) )
     => ( ( hAPP_int_int(div_mod_int(B_71),C_28) = hAPP_int_int(div_mod_int(B_70),C_28) )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_87),B_71)),C_28) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_86),B_70)),C_28) ) ) ) ).

tff(fact_1888_zmod__simps_I1_J,axiom,
    ! [A_85: quickcheck_code_int,C_27: quickcheck_code_int,B_69: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_85),C_27)),B_69)),C_27) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_85),B_69)),C_27) ).

tff(fact_1889_zmod__simps_I1_J,axiom,
    ! [A_85: code_code_numeral,C_27: code_code_numeral,B_69: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_85),C_27)),B_69)),C_27) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_85),B_69)),C_27) ).

tff(fact_1890_zmod__simps_I1_J,axiom,
    ! [A_85: nat,C_27: nat,B_69: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_85),C_27)),B_69)),C_27) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_85),B_69)),C_27) ).

tff(fact_1891_zmod__simps_I1_J,axiom,
    ! [A_85: int,C_27: int,B_69: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_85),C_27)),B_69)),C_27) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_85),B_69)),C_27) ).

tff(fact_1892_zmod__simps_I2_J,axiom,
    ! [A_84: quickcheck_code_int,B_68: quickcheck_code_int,C_26: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_84),hAPP_Q1762011733de_int(div_mo231679042de_int(B_68),C_26))),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_84),B_68)),C_26) ).

tff(fact_1893_zmod__simps_I2_J,axiom,
    ! [A_84: code_code_numeral,B_68: code_code_numeral,C_26: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_84),hAPP_C498520661umeral(div_mo1740067990umeral(B_68),C_26))),C_26) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_84),B_68)),C_26) ).

tff(fact_1894_zmod__simps_I2_J,axiom,
    ! [A_84: nat,B_68: nat,C_26: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_84),hAPP_nat_nat(div_mod_nat(B_68),C_26))),C_26) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_84),B_68)),C_26) ).

tff(fact_1895_zmod__simps_I2_J,axiom,
    ! [A_84: int,B_68: int,C_26: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_84),hAPP_int_int(div_mod_int(B_68),C_26))),C_26) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_84),B_68)),C_26) ).

tff(fact_1896_mod__add__eq,axiom,
    ! [A_83: quickcheck_code_int,B_67: quickcheck_code_int,C_25: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_83),B_67)),C_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_83),C_25)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_67),C_25))),C_25) ).

tff(fact_1897_mod__add__eq,axiom,
    ! [A_83: code_code_numeral,B_67: code_code_numeral,C_25: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_83),B_67)),C_25) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_83),C_25)),hAPP_C498520661umeral(div_mo1740067990umeral(B_67),C_25))),C_25) ).

tff(fact_1898_mod__add__eq,axiom,
    ! [A_83: nat,B_67: nat,C_25: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_83),B_67)),C_25) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_83),C_25)),hAPP_nat_nat(div_mod_nat(B_67),C_25))),C_25) ).

tff(fact_1899_mod__add__eq,axiom,
    ! [A_83: int,B_67: int,C_25: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_83),B_67)),C_25) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_83),C_25)),hAPP_int_int(div_mod_int(B_67),C_25))),C_25) ).

tff(fact_1900_mod__add__left__eq,axiom,
    ! [A_82: quickcheck_code_int,B_66: quickcheck_code_int,C_24: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_82),B_66)),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_82),C_24)),B_66)),C_24) ).

tff(fact_1901_mod__add__left__eq,axiom,
    ! [A_82: code_code_numeral,B_66: code_code_numeral,C_24: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_82),B_66)),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_82),C_24)),B_66)),C_24) ).

tff(fact_1902_mod__add__left__eq,axiom,
    ! [A_82: nat,B_66: nat,C_24: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_82),B_66)),C_24) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_82),C_24)),B_66)),C_24) ).

tff(fact_1903_mod__add__left__eq,axiom,
    ! [A_82: int,B_66: int,C_24: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_82),B_66)),C_24) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_82),C_24)),B_66)),C_24) ).

tff(fact_1904_mod__add__right__eq,axiom,
    ! [A_81: quickcheck_code_int,B_65: quickcheck_code_int,C_23: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_81),B_65)),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_81),hAPP_Q1762011733de_int(div_mo231679042de_int(B_65),C_23))),C_23) ).

tff(fact_1905_mod__add__right__eq,axiom,
    ! [A_81: code_code_numeral,B_65: code_code_numeral,C_23: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_81),B_65)),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_81),hAPP_C498520661umeral(div_mo1740067990umeral(B_65),C_23))),C_23) ).

tff(fact_1906_mod__add__right__eq,axiom,
    ! [A_81: nat,B_65: nat,C_23: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),B_65)),C_23) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),hAPP_nat_nat(div_mod_nat(B_65),C_23))),C_23) ).

tff(fact_1907_mod__add__right__eq,axiom,
    ! [A_81: int,B_65: int,C_23: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_81),B_65)),C_23) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_81),hAPP_int_int(div_mod_int(B_65),C_23))),C_23) ).

tff(fact_1908_mod__add__self1,axiom,
    ! [B_64: quickcheck_code_int,A_80: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_64),A_80)),B_64) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_80),B_64) ).

tff(fact_1909_mod__add__self1,axiom,
    ! [B_64: code_code_numeral,A_80: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_64),A_80)),B_64) = hAPP_C498520661umeral(div_mo1740067990umeral(A_80),B_64) ).

tff(fact_1910_mod__add__self1,axiom,
    ! [B_64: nat,A_80: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_64),A_80)),B_64) = hAPP_nat_nat(div_mod_nat(A_80),B_64) ).

tff(fact_1911_mod__add__self1,axiom,
    ! [B_64: int,A_80: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_64),A_80)),B_64) = hAPP_int_int(div_mod_int(A_80),B_64) ).

tff(fact_1912_mod__add__self2,axiom,
    ! [A_79: quickcheck_code_int,B_63: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_79),B_63)),B_63) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_79),B_63) ).

tff(fact_1913_mod__add__self2,axiom,
    ! [A_79: code_code_numeral,B_63: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_79),B_63)),B_63) = hAPP_C498520661umeral(div_mo1740067990umeral(A_79),B_63) ).

tff(fact_1914_mod__add__self2,axiom,
    ! [A_79: nat,B_63: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),B_63)),B_63) = hAPP_nat_nat(div_mod_nat(A_79),B_63) ).

tff(fact_1915_mod__add__self2,axiom,
    ! [A_79: int,B_63: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_79),B_63)),B_63) = hAPP_int_int(div_mod_int(A_79),B_63) ).

tff(fact_1916_abs__mult__self,axiom,
    ! [A_78: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_78)),hAPP_rat_rat(abs_abs_rat,A_78)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_78),A_78) ).

tff(fact_1917_abs__mult__self,axiom,
    ! [A_78: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_78)),hAPP_real_real(abs_abs_real,A_78)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),A_78) ).

tff(fact_1918_abs__mult__self,axiom,
    ! [A_78: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_78)),hAPP_int_int(abs_abs_int,A_78)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_78),A_78) ).

tff(fact_1919_abs__mult,axiom,
    ! [A_77: rat,B_62: rat] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_77),B_62)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_77)),hAPP_rat_rat(abs_abs_rat,B_62)) ).

tff(fact_1920_abs__mult,axiom,
    ! [A_77: real,B_62: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),B_62)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_77)),hAPP_real_real(abs_abs_real,B_62)) ).

tff(fact_1921_abs__mult,axiom,
    ! [A_77: int,B_62: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_77),B_62)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_77)),hAPP_int_int(abs_abs_int,B_62)) ).

tff(fact_1922_abs__add__abs,axiom,
    ! [A_76: rat,B_61: rat] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_76)),hAPP_rat_rat(abs_abs_rat,B_61))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_76)),hAPP_rat_rat(abs_abs_rat,B_61)) ).

tff(fact_1923_abs__add__abs,axiom,
    ! [A_76: real,B_61: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_76)),hAPP_real_real(abs_abs_real,B_61))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_76)),hAPP_real_real(abs_abs_real,B_61)) ).

tff(fact_1924_abs__add__abs,axiom,
    ! [A_76: int,B_61: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_76)),hAPP_int_int(abs_abs_int,B_61))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_76)),hAPP_int_int(abs_abs_int,B_61)) ).

tff(fact_1925_mod__diff__cong,axiom,
    ! [B_60: int,B_59: int,A_75: int,C_22: int,A_74: int] :
      ( ( hAPP_int_int(div_mod_int(A_75),C_22) = hAPP_int_int(div_mod_int(A_74),C_22) )
     => ( ( hAPP_int_int(div_mod_int(B_60),C_22) = hAPP_int_int(div_mod_int(B_59),C_22) )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_75),B_60)),C_22) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_74),B_59)),C_22) ) ) ) ).

tff(fact_1926_mod__diff__eq,axiom,
    ! [A_73: int,B_58: int,C_21: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_73),B_58)),C_21) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_73),C_21)),hAPP_int_int(div_mod_int(B_58),C_21))),C_21) ).

tff(fact_1927_mod__diff__left__eq,axiom,
    ! [A_72: int,B_57: int,C_20: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_72),B_57)),C_20) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_72),C_20)),B_57)),C_20) ).

tff(fact_1928_mod__diff__right__eq,axiom,
    ! [A_71: int,B_56: int,C_19: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_71),B_56)),C_19) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_71),hAPP_int_int(div_mod_int(B_56),C_19))),C_19) ).

tff(fact_1929_abs__one,axiom,
    hAPP_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

tff(fact_1930_abs__one,axiom,
    hAPP_real_real(abs_abs_real,one_one_real) = one_one_real ).

tff(fact_1931_abs__one,axiom,
    hAPP_int_int(abs_abs_int,one_one_int) = one_one_int ).

tff(fact_1932_abs__minus__commute,axiom,
    ! [A_70: rat,B_55: rat] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_70),B_55)) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_55),A_70)) ).

tff(fact_1933_abs__minus__commute,axiom,
    ! [A_70: real,B_55: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_70),B_55)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_55),A_70)) ).

tff(fact_1934_abs__minus__commute,axiom,
    ! [A_70: int,B_55: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_70),B_55)) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_55),A_70)) ).

tff(fact_1935_floor__minus,axiom,
    ! [X_16: real] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,X_16)) = hAPP_int_int(uminus_uminus_int,archim856651990g_real(X_16)) ).

tff(fact_1936_floor__minus,axiom,
    ! [X_16: rat] : archim791455193or_rat(hAPP_rat_rat(uminus_uminus_rat,X_16)) = hAPP_int_int(uminus_uminus_int,archim1419379755ng_rat(X_16)) ).

tff(fact_1937_ceiling__minus,axiom,
    ! [X_15: real] : archim856651990g_real(hAPP_real_real(uminus_uminus_real,X_15)) = hAPP_int_int(uminus_uminus_int,archim1246769320r_real(X_15)) ).

tff(fact_1938_ceiling__minus,axiom,
    ! [X_15: rat] : archim1419379755ng_rat(hAPP_rat_rat(uminus_uminus_rat,X_15)) = hAPP_int_int(uminus_uminus_int,archim791455193or_rat(X_15)) ).

tff(fact_1939_ceiling__def,axiom,
    ! [X_14: real] : archim856651990g_real(X_14) = hAPP_int_int(uminus_uminus_int,archim1246769320r_real(hAPP_real_real(uminus_uminus_real,X_14))) ).

tff(fact_1940_ceiling__def,axiom,
    ! [X_14: rat] : archim1419379755ng_rat(X_14) = hAPP_int_int(uminus_uminus_int,archim791455193or_rat(hAPP_rat_rat(uminus_uminus_rat,X_14))) ).

tff(fact_1941_neg__equal__zero,axiom,
    ! [Aa: rat] :
      ( ( hAPP_rat_rat(uminus_uminus_rat,Aa) = Aa )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1942_neg__equal__zero,axiom,
    ! [Aa: real] :
      ( ( hAPP_real_real(uminus_uminus_real,Aa) = Aa )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1943_neg__equal__zero,axiom,
    ! [Aa: int] :
      ( ( hAPP_int_int(uminus_uminus_int,Aa) = Aa )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1944_neg__equal__0__iff__equal,axiom,
    ! [Aa: rat] :
      ( ( hAPP_rat_rat(uminus_uminus_rat,Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1945_neg__equal__0__iff__equal,axiom,
    ! [Aa: complex] :
      ( ( hAPP_complex_complex(uminus473333897omplex,Aa) = zero_zero_complex )
    <=> ( Aa = zero_zero_complex ) ) ).

tff(fact_1946_neg__equal__0__iff__equal,axiom,
    ! [Aa: real] :
      ( ( hAPP_real_real(uminus_uminus_real,Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1947_neg__equal__0__iff__equal,axiom,
    ! [Aa: int] :
      ( ( hAPP_int_int(uminus_uminus_int,Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1948_equal__neg__zero,axiom,
    ! [Aa: rat] :
      ( ( Aa = hAPP_rat_rat(uminus_uminus_rat,Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_1949_equal__neg__zero,axiom,
    ! [Aa: real] :
      ( ( Aa = hAPP_real_real(uminus_uminus_real,Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_1950_equal__neg__zero,axiom,
    ! [Aa: int] :
      ( ( Aa = hAPP_int_int(uminus_uminus_int,Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_1951_neg__0__equal__iff__equal,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = hAPP_rat_rat(uminus_uminus_rat,Aa) )
    <=> ( zero_zero_rat = Aa ) ) ).

tff(fact_1952_neg__0__equal__iff__equal,axiom,
    ! [Aa: complex] :
      ( ( zero_zero_complex = hAPP_complex_complex(uminus473333897omplex,Aa) )
    <=> ( zero_zero_complex = Aa ) ) ).

tff(fact_1953_neg__0__equal__iff__equal,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = hAPP_real_real(uminus_uminus_real,Aa) )
    <=> ( zero_zero_real = Aa ) ) ).

tff(fact_1954_neg__0__equal__iff__equal,axiom,
    ! [Aa: int] :
      ( ( zero_zero_int = hAPP_int_int(uminus_uminus_int,Aa) )
    <=> ( zero_zero_int = Aa ) ) ).

tff(fact_1955_minus__zero,axiom,
    hAPP_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

tff(fact_1956_minus__zero,axiom,
    hAPP_complex_complex(uminus473333897omplex,zero_zero_complex) = zero_zero_complex ).

tff(fact_1957_minus__zero,axiom,
    hAPP_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

tff(fact_1958_minus__zero,axiom,
    hAPP_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

tff(fact_1959_compl__le__swap2,axiom,
    ! [Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2)),X_1))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,X_1)),Y_2)) ) ).

tff(fact_1960_compl__le__swap2,axiom,
    ! [Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2)),X_1))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,X_1)),Y_2)) ) ).

tff(fact_1961_compl__le__swap1,axiom,
    ! [Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),hAPP_f1805168059t_bool(uminus407446779t_bool,X_1)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2))) ) ).

tff(fact_1962_compl__le__swap1,axiom,
    ! [Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),hAPP_f800510211t_bool(uminus1024588959t_bool,X_1)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2))) ) ).

tff(fact_1963_compl__mono,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2)),hAPP_f1805168059t_bool(uminus407446779t_bool,X_1))) ) ).

tff(fact_1964_compl__mono,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2)),hAPP_f800510211t_bool(uminus1024588959t_bool,X_1))) ) ).

tff(fact_1965_compl__le__compl__iff,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,X_1)),hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2)))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_2),X_1)) ) ).

tff(fact_1966_compl__le__compl__iff,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,X_1)),hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2)))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_1)) ) ).

tff(fact_1967_le__imp__neg__le,axiom,
    ! [A_69: rat,B_54: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_69),B_54))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,B_54)),hAPP_rat_rat(uminus_uminus_rat,A_69))) ) ).

tff(fact_1968_le__imp__neg__le,axiom,
    ! [A_69: real,B_54: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_69),B_54))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,B_54)),hAPP_real_real(uminus_uminus_real,A_69))) ) ).

tff(fact_1969_le__imp__neg__le,axiom,
    ! [A_69: int,B_54: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_69),B_54))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,B_54)),hAPP_int_int(uminus_uminus_int,A_69))) ) ).

tff(fact_1970_neg__le__iff__le,axiom,
    ! [Ba: rat,Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,Ba)),hAPP_rat_rat(uminus_uminus_rat,Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),Ba)) ) ).

tff(fact_1971_neg__le__iff__le,axiom,
    ! [Ba: real,Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Ba)),hAPP_real_real(uminus_uminus_real,Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).

tff(fact_1972_neg__le__iff__le,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,Ba)),hAPP_int_int(uminus_uminus_int,Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).

tff(fact_1973_minus__le__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),Ba))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,Ba)),Aa)) ) ).

tff(fact_1974_minus__le__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Aa)),Ba))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Ba)),Aa)) ) ).

tff(fact_1975_minus__le__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,Aa)),Ba))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,Ba)),Aa)) ) ).

tff(fact_1976_le__minus__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),hAPP_rat_rat(uminus_uminus_rat,Ba)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ba),hAPP_rat_rat(uminus_uminus_rat,Aa))) ) ).

tff(fact_1977_le__minus__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),hAPP_real_real(uminus_uminus_real,Ba)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),hAPP_real_real(uminus_uminus_real,Aa))) ) ).

tff(fact_1978_le__minus__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),hAPP_int_int(uminus_uminus_int,Ba)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),hAPP_int_int(uminus_uminus_int,Aa))) ) ).

tff(fact_1979_compl__less__swap2,axiom,
    ! [Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2)),X_1))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,X_1)),Y_2)) ) ).

tff(fact_1980_compl__less__swap2,axiom,
    ! [Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2)),X_1))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,X_1)),Y_2)) ) ).

tff(fact_1981_compl__less__swap1,axiom,
    ! [Y_2: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),hAPP_f1805168059t_bool(uminus407446779t_bool,X_1)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2))) ) ).

tff(fact_1982_compl__less__swap1,axiom,
    ! [Y_2: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),hAPP_f800510211t_bool(uminus1024588959t_bool,X_1)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2))) ) ).

tff(fact_1983_compl__less__compl__iff,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,hAPP_f1805168059t_bool(uminus407446779t_bool,X_1)),hAPP_f1805168059t_bool(uminus407446779t_bool,Y_2)))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Y_2),X_1)) ) ).

tff(fact_1984_compl__less__compl__iff,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,hAPP_f800510211t_bool(uminus1024588959t_bool,X_1)),hAPP_f800510211t_bool(uminus1024588959t_bool,Y_2)))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_2),X_1)) ) ).

tff(fact_1985_neg__less__iff__less,axiom,
    ! [Ba: rat,Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(uminus_uminus_rat,Ba)),hAPP_rat_rat(uminus_uminus_rat,Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),Ba)) ) ).

tff(fact_1986_neg__less__iff__less,axiom,
    ! [Ba: real,Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Ba)),hAPP_real_real(uminus_uminus_real,Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).

tff(fact_1987_neg__less__iff__less,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,Ba)),hAPP_int_int(uminus_uminus_int,Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).

tff(fact_1988_minus__less__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),Ba))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(uminus_uminus_rat,Ba)),Aa)) ) ).

tff(fact_1989_minus__less__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Aa)),Ba))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Ba)),Aa)) ) ).

tff(fact_1990_minus__less__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,Aa)),Ba))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,Ba)),Aa)) ) ).

tff(fact_1991_less__minus__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),hAPP_rat_rat(uminus_uminus_rat,Ba)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ba),hAPP_rat_rat(uminus_uminus_rat,Aa))) ) ).

tff(fact_1992_less__minus__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),hAPP_real_real(uminus_uminus_real,Ba)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),hAPP_real_real(uminus_uminus_real,Aa))) ) ).

tff(fact_1993_less__minus__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(uminus_uminus_int,Ba)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),hAPP_int_int(uminus_uminus_int,Aa))) ) ).

tff(fact_1994_square__eq__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),Aa) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = hAPP_rat_rat(uminus_uminus_rat,Ba) ) ) ) ).

tff(fact_1995_square__eq__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),Aa) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = hAPP_complex_complex(uminus473333897omplex,Ba) ) ) ) ).

tff(fact_1996_square__eq__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),Aa) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = hAPP_real_real(uminus_uminus_real,Ba) ) ) ) ).

tff(fact_1997_square__eq__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Aa) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = hAPP_int_int(uminus_uminus_int,Ba) ) ) ) ).

tff(fact_1998_minus__mult__minus,axiom,
    ! [A_68: rat,B_53: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(uminus_uminus_rat,A_68)),hAPP_rat_rat(uminus_uminus_rat,B_53)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_68),B_53) ).

tff(fact_1999_minus__mult__minus,axiom,
    ! [A_68: complex,B_53: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,A_68)),hAPP_complex_complex(uminus473333897omplex,B_53)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_68),B_53) ).

tff(fact_2000_minus__mult__minus,axiom,
    ! [A_68: real,B_53: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,A_68)),hAPP_real_real(uminus_uminus_real,B_53)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_68),B_53) ).

tff(fact_2001_minus__mult__minus,axiom,
    ! [A_68: int,B_53: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,A_68)),hAPP_int_int(uminus_uminus_int,B_53)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_68),B_53) ).

tff(fact_2002_minus__mult__commute,axiom,
    ! [A_67: rat,B_52: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(uminus_uminus_rat,A_67)),B_52) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_67),hAPP_rat_rat(uminus_uminus_rat,B_52)) ).

tff(fact_2003_minus__mult__commute,axiom,
    ! [A_67: complex,B_52: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,A_67)),B_52) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_67),hAPP_complex_complex(uminus473333897omplex,B_52)) ).

tff(fact_2004_minus__mult__commute,axiom,
    ! [A_67: real,B_52: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,A_67)),B_52) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_67),hAPP_real_real(uminus_uminus_real,B_52)) ).

tff(fact_2005_minus__mult__commute,axiom,
    ! [A_67: int,B_52: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,A_67)),B_52) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_67),hAPP_int_int(uminus_uminus_int,B_52)) ).

tff(fact_2006_mult__left_Ominus,axiom,
    ! [X_13: complex,Y_11: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,X_13)),Y_11) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_13),Y_11)) ).

tff(fact_2007_mult__left_Ominus,axiom,
    ! [X_13: real,Y_11: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,X_13)),Y_11) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_13),Y_11)) ).

tff(fact_2008_mult_Ominus__left,axiom,
    ! [A_66: complex,B_51: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,A_66)),B_51) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_66),B_51)) ).

tff(fact_2009_mult_Ominus__left,axiom,
    ! [A_66: real,B_51: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,A_66)),B_51) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_66),B_51)) ).

tff(fact_2010_mult__right_Ominus,axiom,
    ! [Xa_3: complex,X_12: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),hAPP_complex_complex(uminus473333897omplex,X_12)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),X_12)) ).

tff(fact_2011_mult__right_Ominus,axiom,
    ! [Xa_3: real,X_12: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),hAPP_real_real(uminus_uminus_real,X_12)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),X_12)) ).

tff(fact_2012_mult_Ominus__right,axiom,
    ! [A_65: complex,B_50: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_65),hAPP_complex_complex(uminus473333897omplex,B_50)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_65),B_50)) ).

tff(fact_2013_mult_Ominus__right,axiom,
    ! [A_65: real,B_50: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_65),hAPP_real_real(uminus_uminus_real,B_50)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_65),B_50)) ).

tff(fact_2014_minus__mult__left,axiom,
    ! [A_64: rat,B_49: rat] : hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_64),B_49)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(uminus_uminus_rat,A_64)),B_49) ).

tff(fact_2015_minus__mult__left,axiom,
    ! [A_64: complex,B_49: complex] : hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_64),B_49)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,A_64)),B_49) ).

tff(fact_2016_minus__mult__left,axiom,
    ! [A_64: real,B_49: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_64),B_49)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,A_64)),B_49) ).

tff(fact_2017_minus__mult__left,axiom,
    ! [A_64: int,B_49: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_64),B_49)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,A_64)),B_49) ).

tff(fact_2018_minus__mult__right,axiom,
    ! [A_63: rat,B_48: rat] : hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_63),B_48)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_63),hAPP_rat_rat(uminus_uminus_rat,B_48)) ).

tff(fact_2019_minus__mult__right,axiom,
    ! [A_63: complex,B_48: complex] : hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_63),B_48)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_63),hAPP_complex_complex(uminus473333897omplex,B_48)) ).

tff(fact_2020_minus__mult__right,axiom,
    ! [A_63: real,B_48: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_63),B_48)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_63),hAPP_real_real(uminus_uminus_real,B_48)) ).

tff(fact_2021_minus__mult__right,axiom,
    ! [A_63: int,B_48: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_63),B_48)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_63),hAPP_int_int(uminus_uminus_int,B_48)) ).

tff(fact_2022_minus__add__cancel,axiom,
    ! [A_62: rat,B_47: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(uminus_uminus_rat,A_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_62),B_47)) = B_47 ).

tff(fact_2023_minus__add__cancel,axiom,
    ! [A_62: complex,B_47: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(uminus473333897omplex,A_62)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_62),B_47)) = B_47 ).

tff(fact_2024_minus__add__cancel,axiom,
    ! [A_62: real,B_47: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,A_62)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_62),B_47)) = B_47 ).

tff(fact_2025_minus__add__cancel,axiom,
    ! [A_62: int,B_47: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,A_62)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_62),B_47)) = B_47 ).

tff(fact_2026_add__minus__cancel,axiom,
    ! [A_61: rat,B_46: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_61),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(uminus_uminus_rat,A_61)),B_46)) = B_46 ).

tff(fact_2027_add__minus__cancel,axiom,
    ! [A_61: complex,B_46: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_61),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(uminus473333897omplex,A_61)),B_46)) = B_46 ).

tff(fact_2028_add__minus__cancel,axiom,
    ! [A_61: real,B_46: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_61),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,A_61)),B_46)) = B_46 ).

tff(fact_2029_add__minus__cancel,axiom,
    ! [A_61: int,B_46: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_61),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,A_61)),B_46)) = B_46 ).

tff(fact_2030_minus__add,axiom,
    ! [A_60: rat,B_45: rat] : hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_60),B_45)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(uminus_uminus_rat,B_45)),hAPP_rat_rat(uminus_uminus_rat,A_60)) ).

tff(fact_2031_minus__add,axiom,
    ! [A_60: complex,B_45: complex] : hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_60),B_45)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(uminus473333897omplex,B_45)),hAPP_complex_complex(uminus473333897omplex,A_60)) ).

tff(fact_2032_minus__add,axiom,
    ! [A_60: real,B_45: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_60),B_45)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,B_45)),hAPP_real_real(uminus_uminus_real,A_60)) ).

tff(fact_2033_minus__add,axiom,
    ! [A_60: int,B_45: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_60),B_45)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,B_45)),hAPP_int_int(uminus_uminus_int,A_60)) ).

tff(fact_2034_minus__add__distrib,axiom,
    ! [A_59: rat,B_44: rat] : hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_59),B_44)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(uminus_uminus_rat,A_59)),hAPP_rat_rat(uminus_uminus_rat,B_44)) ).

tff(fact_2035_minus__add__distrib,axiom,
    ! [A_59: complex,B_44: complex] : hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_59),B_44)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(uminus473333897omplex,A_59)),hAPP_complex_complex(uminus473333897omplex,B_44)) ).

tff(fact_2036_minus__add__distrib,axiom,
    ! [A_59: real,B_44: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_59),B_44)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,A_59)),hAPP_real_real(uminus_uminus_real,B_44)) ).

tff(fact_2037_minus__add__distrib,axiom,
    ! [A_59: int,B_44: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_59),B_44)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,A_59)),hAPP_int_int(uminus_uminus_int,B_44)) ).

tff(fact_2038_zmod__self,axiom,
    ! [A_2: int] : hAPP_int_int(div_mod_int(A_2),A_2) = zero_zero_int ).

tff(fact_2039_zmod__zero,axiom,
    ! [B_1: int] : hAPP_int_int(div_mod_int(zero_zero_int),B_1) = zero_zero_int ).

tff(fact_2040_minus__diff__minus,axiom,
    ! [A_58: rat,B_43: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(uminus_uminus_rat,A_58)),hAPP_rat_rat(uminus_uminus_rat,B_43)) = hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_58),B_43)) ).

tff(fact_2041_minus__diff__minus,axiom,
    ! [A_58: complex,B_43: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(uminus473333897omplex,A_58)),hAPP_complex_complex(uminus473333897omplex,B_43)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_58),B_43)) ).

tff(fact_2042_minus__diff__minus,axiom,
    ! [A_58: real,B_43: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,A_58)),hAPP_real_real(uminus_uminus_real,B_43)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_58),B_43)) ).

tff(fact_2043_minus__diff__minus,axiom,
    ! [A_58: int,B_43: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,A_58)),hAPP_int_int(uminus_uminus_int,B_43)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_58),B_43)) ).

tff(fact_2044_minus__diff__eq,axiom,
    ! [A_57: rat,B_42: rat] : hAPP_rat_rat(uminus_uminus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_57),B_42)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_42),A_57) ).

tff(fact_2045_minus__diff__eq,axiom,
    ! [A_57: complex,B_42: complex] : hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_57),B_42)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_42),A_57) ).

tff(fact_2046_minus__diff__eq,axiom,
    ! [A_57: real,B_42: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_57),B_42)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_42),A_57) ).

tff(fact_2047_minus__diff__eq,axiom,
    ! [A_57: int,B_42: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_57),B_42)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_42),A_57) ).

tff(fact_2048_zmod__simps_I3_J,axiom,
    ! [A_2: int,B_1: int,C_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(div_mod_int(B_1),C_1))),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)),C_1) ).

tff(fact_2049_zmod__zmult1__eq,axiom,
    ! [A_2: int,B_1: int,C_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(div_mod_int(B_1),C_1))),C_1) ).

tff(fact_2050_zminus__0,axiom,
    hAPP_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

tff(fact_2051_zdiff__zmod__left,axiom,
    ! [X: int,M_1: int,Y: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(X),M_1)),Y)),M_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M_1) ).

tff(fact_2052_zdiff__zmod__right,axiom,
    ! [X: int,Y: int,M_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(div_mod_int(Y),M_1))),M_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M_1) ).

tff(fact_2053_zmult__zminus,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z_1)),W) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W)) ).

tff(fact_2054_rel__simps_I7_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).

tff(fact_2055_minus__numeral__code_I5_J,axiom,
    ! [W: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W)) ).

tff(fact_2056_rel__simps_I24_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).

tff(fact_2057_zminus__zadd__distrib,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),hAPP_int_int(uminus_uminus_int,W)) ).

tff(fact_2058_abs__number__of,axiom,
    ! [X_11: int] :
      ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_11)),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(number_number_of_rat,X_11)) = hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(number_number_of_rat,X_11)) ) )
      & ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_11)),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(number_number_of_rat,X_11)) = hAPP_int_rat(number_number_of_rat,X_11) ) ) ) ).

tff(fact_2059_abs__number__of,axiom,
    ! [X_11: int] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_11)),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,hAPP_int_real(number267125858f_real,X_11)) = hAPP_real_real(uminus_uminus_real,hAPP_int_real(number267125858f_real,X_11)) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_11)),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,hAPP_int_real(number267125858f_real,X_11)) = hAPP_int_real(number267125858f_real,X_11) ) ) ) ).

tff(fact_2060_abs__number__of,axiom,
    ! [X_11: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_11)),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,hAPP_int_int(number_number_of_int,X_11)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,X_11)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_11)),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,hAPP_int_int(number_number_of_int,X_11)) = hAPP_int_int(number_number_of_int,X_11) ) ) ) ).

tff(fact_2061_Ints__0,axiom,
    hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,zero_zero_real),ring_1_Ints_real)) ).

tff(fact_2062_Ints__0,axiom,
    hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,zero_zero_int),ring_1_Ints_int)) ).

tff(fact_2063_Ints__0,axiom,
    hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,zero_zero_rat),ring_1_Ints_rat)) ).

tff(fact_2064_Ints__0,axiom,
    hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,zero_zero_complex),ring_1_Ints_complex)) ).

tff(fact_2065_Ints__mult,axiom,
    ! [Ba: real,Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Ba),ring_1_Ints_real))
       => hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Aa),Ba)),ring_1_Ints_real)) ) ) ).

tff(fact_2066_Ints__mult,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),ring_1_Ints_int))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba)),ring_1_Ints_int)) ) ) ).

tff(fact_2067_Ints__mult,axiom,
    ! [Ba: rat,Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Ba),ring_1_Ints_rat))
       => hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Aa),Ba)),ring_1_Ints_rat)) ) ) ).

tff(fact_2068_Ints__mult,axiom,
    ! [Ba: complex,Aa: complex] :
      ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Aa),ring_1_Ints_complex))
     => ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Ba),ring_1_Ints_complex))
       => hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Aa),Ba)),ring_1_Ints_complex)) ) ) ).

tff(fact_2069_Ints__add,axiom,
    ! [Ba: real,Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Ba),ring_1_Ints_real))
       => hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Ba)),ring_1_Ints_real)) ) ) ).

tff(fact_2070_Ints__add,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),ring_1_Ints_int))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)),ring_1_Ints_int)) ) ) ).

tff(fact_2071_Ints__add,axiom,
    ! [Ba: rat,Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Ba),ring_1_Ints_rat))
       => hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Ba)),ring_1_Ints_rat)) ) ) ).

tff(fact_2072_Ints__add,axiom,
    ! [Ba: complex,Aa: complex] :
      ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Aa),ring_1_Ints_complex))
     => ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Ba),ring_1_Ints_complex))
       => hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Aa),Ba)),ring_1_Ints_complex)) ) ) ).

tff(fact_2073_Ints__number__of,axiom,
    ! [W_1: int] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(number267125858f_real,W_1)),ring_1_Ints_real)) ).

tff(fact_2074_Ints__number__of,axiom,
    ! [W_1: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(number_number_of_int,W_1)),ring_1_Ints_int)) ).

tff(fact_2075_Ints__number__of,axiom,
    ! [W_1: int] : hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,hAPP_int_rat(number_number_of_rat,W_1)),ring_1_Ints_rat)) ).

tff(fact_2076_Ints__number__of,axiom,
    ! [W_1: int] : hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,hAPP_int_complex(number528085621omplex,W_1)),ring_1_Ints_complex)) ).

tff(fact_2077_Ints__1,axiom,
    hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,one_one_real),ring_1_Ints_real)) ).

tff(fact_2078_Ints__1,axiom,
    hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,one_one_int),ring_1_Ints_int)) ).

tff(fact_2079_Ints__1,axiom,
    hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,one_one_rat),ring_1_Ints_rat)) ).

tff(fact_2080_Ints__1,axiom,
    hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,one_one_complex),ring_1_Ints_complex)) ).

tff(fact_2081_Ints__diff,axiom,
    ! [Ba: real,Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Ba),ring_1_Ints_real))
       => hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),Ba)),ring_1_Ints_real)) ) ) ).

tff(fact_2082_Ints__diff,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),ring_1_Ints_int))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),ring_1_Ints_int)) ) ) ).

tff(fact_2083_Ints__diff,axiom,
    ! [Ba: rat,Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Ba),ring_1_Ints_rat))
       => hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),Ba)),ring_1_Ints_rat)) ) ) ).

tff(fact_2084_Ints__diff,axiom,
    ! [Ba: complex,Aa: complex] :
      ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Aa),ring_1_Ints_complex))
     => ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Ba),ring_1_Ints_complex))
       => hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Aa),Ba)),ring_1_Ints_complex)) ) ) ).

tff(fact_2085_mod__mult__self1__is__0,axiom,
    ! [B_41: quickcheck_code_int,A_56: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_41),A_56)),B_41) = zero_z891286103de_int ).

tff(fact_2086_mod__mult__self1__is__0,axiom,
    ! [B_41: code_code_numeral,A_56: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_41),A_56)),B_41) = zero_z126310315umeral ).

tff(fact_2087_mod__mult__self1__is__0,axiom,
    ! [B_41: nat,A_56: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_41),A_56)),B_41) = zero_zero_nat ).

tff(fact_2088_mod__mult__self1__is__0,axiom,
    ! [B_41: int,A_56: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_41),A_56)),B_41) = zero_zero_int ).

tff(fact_2089_mod__mult__self2__is__0,axiom,
    ! [A_55: quickcheck_code_int,B_40: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_55),B_40)),B_40) = zero_z891286103de_int ).

tff(fact_2090_mod__mult__self2__is__0,axiom,
    ! [A_55: code_code_numeral,B_40: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_55),B_40)),B_40) = zero_z126310315umeral ).

tff(fact_2091_mod__mult__self2__is__0,axiom,
    ! [A_55: nat,B_40: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_55),B_40)),B_40) = zero_zero_nat ).

tff(fact_2092_mod__mult__self2__is__0,axiom,
    ! [A_55: int,B_40: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_55),B_40)),B_40) = zero_zero_int ).

tff(fact_2093_abs__of__nonneg,axiom,
    ! [A_54: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_54))
     => ( hAPP_rat_rat(abs_abs_rat,A_54) = A_54 ) ) ).

tff(fact_2094_abs__of__nonneg,axiom,
    ! [A_54: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_54))
     => ( hAPP_real_real(abs_abs_real,A_54) = A_54 ) ) ).

tff(fact_2095_abs__of__nonneg,axiom,
    ! [A_54: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_54))
     => ( hAPP_int_int(abs_abs_int,A_54) = A_54 ) ) ).

tff(fact_2096_abs__le__zero__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,Aa)),zero_zero_rat))
    <=> ( Aa = zero_zero_rat ) ) ).

tff(fact_2097_abs__le__zero__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Aa)),zero_zero_real))
    <=> ( Aa = zero_zero_real ) ) ).

tff(fact_2098_abs__le__zero__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,Aa)),zero_zero_int))
    <=> ( Aa = zero_zero_int ) ) ).

tff(fact_2099_abs__ge__zero,axiom,
    ! [A_53: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(abs_abs_rat,A_53))) ).

tff(fact_2100_abs__ge__zero,axiom,
    ! [A_53: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(abs_abs_real,A_53))) ).

tff(fact_2101_abs__ge__zero,axiom,
    ! [A_53: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(abs_abs_int,A_53))) ).

tff(fact_2102_abs__of__pos,axiom,
    ! [A_52: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_52))
     => ( hAPP_rat_rat(abs_abs_rat,A_52) = A_52 ) ) ).

tff(fact_2103_abs__of__pos,axiom,
    ! [A_52: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_52))
     => ( hAPP_real_real(abs_abs_real,A_52) = A_52 ) ) ).

tff(fact_2104_abs__of__pos,axiom,
    ! [A_52: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_52))
     => ( hAPP_int_int(abs_abs_int,A_52) = A_52 ) ) ).

tff(fact_2105_zero__less__abs__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(abs_abs_rat,Aa)))
    <=> ( Aa != zero_zero_rat ) ) ).

tff(fact_2106_zero__less__abs__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,Aa)))
    <=> ( Aa != zero_zero_real ) ) ).

tff(fact_2107_zero__less__abs__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(abs_abs_int,Aa)))
    <=> ( Aa != zero_zero_int ) ) ).

tff(fact_2108_abs__not__less__zero,axiom,
    ! [A_51: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,A_51)),zero_zero_rat)) ).

tff(fact_2109_abs__not__less__zero,axiom,
    ! [A_51: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,A_51)),zero_zero_real)) ).

tff(fact_2110_abs__not__less__zero,axiom,
    ! [A_51: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,A_51)),zero_zero_int)) ).

tff(fact_2111_mod__by__1,axiom,
    ! [A_50: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),one_on1684967323de_int) = zero_z891286103de_int ).

tff(fact_2112_mod__by__1,axiom,
    ! [A_50: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(A_50),one_on1645066479umeral) = zero_z126310315umeral ).

tff(fact_2113_mod__by__1,axiom,
    ! [A_50: nat] : hAPP_nat_nat(div_mod_nat(A_50),one_one_nat) = zero_zero_nat ).

tff(fact_2114_mod__by__1,axiom,
    ! [A_50: int] : hAPP_int_int(div_mod_int(A_50),one_one_int) = zero_zero_int ).

tff(fact_2115_abs__mult__less,axiom,
    ! [B_39: rat,D_6: rat,A_49: rat,C_18: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,A_49)),C_18))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,B_39)),D_6))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_49)),hAPP_rat_rat(abs_abs_rat,B_39))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_18),D_6))) ) ) ).

tff(fact_2116_abs__mult__less,axiom,
    ! [B_39: real,D_6: real,A_49: real,C_18: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,A_49)),C_18))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,B_39)),D_6))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_49)),hAPP_real_real(abs_abs_real,B_39))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_18),D_6))) ) ) ).

tff(fact_2117_abs__mult__less,axiom,
    ! [B_39: int,D_6: int,A_49: int,C_18: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,A_49)),C_18))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,B_39)),D_6))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_49)),hAPP_int_int(abs_abs_int,B_39))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_18),D_6))) ) ) ).

tff(fact_2118_mod__mult__self1,axiom,
    ! [A_48: quickcheck_code_int,C_17: quickcheck_code_int,B_38: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_48),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_17),B_38))),B_38) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_48),B_38) ).

tff(fact_2119_mod__mult__self1,axiom,
    ! [A_48: code_code_numeral,C_17: code_code_numeral,B_38: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_48),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_17),B_38))),B_38) = hAPP_C498520661umeral(div_mo1740067990umeral(A_48),B_38) ).

tff(fact_2120_mod__mult__self1,axiom,
    ! [A_48: nat,C_17: nat,B_38: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_17),B_38))),B_38) = hAPP_nat_nat(div_mod_nat(A_48),B_38) ).

tff(fact_2121_mod__mult__self1,axiom,
    ! [A_48: int,C_17: int,B_38: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_48),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_17),B_38))),B_38) = hAPP_int_int(div_mod_int(A_48),B_38) ).

tff(fact_2122_mod__mult__self2,axiom,
    ! [A_47: quickcheck_code_int,B_37: quickcheck_code_int,C_16: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_47),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_37),C_16))),B_37) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_47),B_37) ).

tff(fact_2123_mod__mult__self2,axiom,
    ! [A_47: code_code_numeral,B_37: code_code_numeral,C_16: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_47),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_37),C_16))),B_37) = hAPP_C498520661umeral(div_mo1740067990umeral(A_47),B_37) ).

tff(fact_2124_mod__mult__self2,axiom,
    ! [A_47: nat,B_37: nat,C_16: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_37),C_16))),B_37) = hAPP_nat_nat(div_mod_nat(A_47),B_37) ).

tff(fact_2125_mod__mult__self2,axiom,
    ! [A_47: int,B_37: int,C_16: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_47),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_37),C_16))),B_37) = hAPP_int_int(div_mod_int(A_47),B_37) ).

tff(fact_2126_abs__triangle__ineq,axiom,
    ! [A_46: rat,B_36: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_46),B_36))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_46)),hAPP_rat_rat(abs_abs_rat,B_36)))) ).

tff(fact_2127_abs__triangle__ineq,axiom,
    ! [A_46: real,B_36: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_46),B_36))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_46)),hAPP_real_real(abs_abs_real,B_36)))) ).

tff(fact_2128_abs__triangle__ineq,axiom,
    ! [A_46: int,B_36: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),B_36))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_46)),hAPP_int_int(abs_abs_int,B_36)))) ).

tff(fact_2129_abs__triangle__ineq3,axiom,
    ! [A_45: rat,B_35: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_45)),hAPP_rat_rat(abs_abs_rat,B_35)))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_45),B_35)))) ).

tff(fact_2130_abs__triangle__ineq3,axiom,
    ! [A_45: real,B_35: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(abs_abs_real,A_45)),hAPP_real_real(abs_abs_real,B_35)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_45),B_35)))) ).

tff(fact_2131_abs__triangle__ineq3,axiom,
    ! [A_45: int,B_35: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_45)),hAPP_int_int(abs_abs_int,B_35)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_45),B_35)))) ).

tff(fact_2132_abs__triangle__ineq2,axiom,
    ! [A_44: rat,B_34: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_44)),hAPP_rat_rat(abs_abs_rat,B_34))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_44),B_34)))) ).

tff(fact_2133_abs__triangle__ineq2,axiom,
    ! [A_44: real,B_34: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(abs_abs_real,A_44)),hAPP_real_real(abs_abs_real,B_34))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_44),B_34)))) ).

tff(fact_2134_abs__triangle__ineq2,axiom,
    ! [A_44: int,B_34: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_44)),hAPP_int_int(abs_abs_int,B_34))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_44),B_34)))) ).

tff(fact_2135_abs__triangle__ineq2__sym,axiom,
    ! [A_43: rat,B_33: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_43)),hAPP_rat_rat(abs_abs_rat,B_33))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_33),A_43)))) ).

tff(fact_2136_abs__triangle__ineq2__sym,axiom,
    ! [A_43: real,B_33: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(abs_abs_real,A_43)),hAPP_real_real(abs_abs_real,B_33))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_33),A_43)))) ).

tff(fact_2137_abs__triangle__ineq2__sym,axiom,
    ! [A_43: int,B_33: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_43)),hAPP_int_int(abs_abs_int,B_33))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_33),A_43)))) ).

tff(fact_2138_abs__diff__less__iff,axiom,
    ! [X_1: rat,Aa: rat,R_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_1),Aa))),R_2))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Aa),R_2)),X_1))
        & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),R_2))) ) ) ).

tff(fact_2139_abs__diff__less__iff,axiom,
    ! [X_1: real,Aa: real,R_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Aa))),R_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Aa),R_2)),X_1))
        & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),R_2))) ) ) ).

tff(fact_2140_abs__diff__less__iff,axiom,
    ! [X_1: int,Aa: int,R_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Aa))),R_2))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),R_2)),X_1))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),R_2))) ) ) ).

tff(fact_2141_d22set__le__swap,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(d22set,Aa))) ) ).

tff(fact_2142_minus__le__self__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),Aa))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Aa)) ) ).

tff(fact_2143_minus__le__self__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Aa)),Aa))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa)) ) ).

tff(fact_2144_minus__le__self__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,Aa)),Aa))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).

tff(fact_2145_neg__le__0__iff__le,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Aa)) ) ).

tff(fact_2146_neg__le__0__iff__le,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Aa)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa)) ) ).

tff(fact_2147_neg__le__0__iff__le,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,Aa)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).

tff(fact_2148_le__minus__self__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),hAPP_rat_rat(uminus_uminus_rat,Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),zero_zero_rat)) ) ).

tff(fact_2149_le__minus__self__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),hAPP_real_real(uminus_uminus_real,Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real)) ) ).

tff(fact_2150_le__minus__self__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),hAPP_int_int(uminus_uminus_int,Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ).

tff(fact_2151_neg__0__le__iff__le,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(uminus_uminus_rat,Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Aa),zero_zero_rat)) ) ).

tff(fact_2152_neg__0__le__iff__le,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(uminus_uminus_real,Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real)) ) ).

tff(fact_2153_neg__0__le__iff__le,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ).

tff(fact_2154_d22set__le,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(d22set,Aa)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa)) ) ).

tff(fact_2155_neg__less__nonneg,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),Aa))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Aa)) ) ).

tff(fact_2156_neg__less__nonneg,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Aa)),Aa))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Aa)) ) ).

tff(fact_2157_neg__less__nonneg,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,Aa)),Aa))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).

tff(fact_2158_neg__less__0__iff__less,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(uminus_uminus_rat,Aa)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Aa)) ) ).

tff(fact_2159_neg__less__0__iff__less,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Aa)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Aa)) ) ).

tff(fact_2160_neg__less__0__iff__less,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,Aa)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).

tff(fact_2161_less__minus__self__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),hAPP_rat_rat(uminus_uminus_rat,Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),zero_zero_rat)) ) ).

tff(fact_2162_less__minus__self__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),hAPP_real_real(uminus_uminus_real,Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),zero_zero_real)) ) ).

tff(fact_2163_less__minus__self__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(uminus_uminus_int,Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).

tff(fact_2164_neg__0__less__iff__less,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(uminus_uminus_rat,Aa)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Aa),zero_zero_rat)) ) ).

tff(fact_2165_neg__0__less__iff__less,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(uminus_uminus_real,Aa)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),zero_zero_real)) ) ).

tff(fact_2166_neg__0__less__iff__less,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,Aa)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).

tff(fact_2167_right__minus,axiom,
    ! [A_42: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_42),hAPP_rat_rat(uminus_uminus_rat,A_42)) = zero_zero_rat ).

tff(fact_2168_right__minus,axiom,
    ! [A_42: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_42),hAPP_complex_complex(uminus473333897omplex,A_42)) = zero_zero_complex ).

tff(fact_2169_right__minus,axiom,
    ! [A_42: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_42),hAPP_real_real(uminus_uminus_real,A_42)) = zero_zero_real ).

tff(fact_2170_right__minus,axiom,
    ! [A_42: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_42),hAPP_int_int(uminus_uminus_int,A_42)) = zero_zero_int ).

tff(fact_2171_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = hAPP_rat_rat(uminus_uminus_rat,Ba) )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Ba) = zero_zero_rat ) ) ).

tff(fact_2172_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( Aa = hAPP_complex_complex(uminus473333897omplex,Ba) )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Aa),Ba) = zero_zero_complex ) ) ).

tff(fact_2173_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = hAPP_real_real(uminus_uminus_real,Ba) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Ba) = zero_zero_real ) ) ).

tff(fact_2174_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = hAPP_int_int(uminus_uminus_int,Ba) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba) = zero_zero_int ) ) ).

tff(fact_2175_left__minus,axiom,
    ! [A_41: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(uminus_uminus_rat,A_41)),A_41) = zero_zero_rat ).

tff(fact_2176_left__minus,axiom,
    ! [A_41: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(uminus473333897omplex,A_41)),A_41) = zero_zero_complex ).

tff(fact_2177_left__minus,axiom,
    ! [A_41: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,A_41)),A_41) = zero_zero_real ).

tff(fact_2178_left__minus,axiom,
    ! [A_41: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,A_41)),A_41) = zero_zero_int ).

tff(fact_2179_ab__left__minus,axiom,
    ! [A_40: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(uminus_uminus_rat,A_40)),A_40) = zero_zero_rat ).

tff(fact_2180_ab__left__minus,axiom,
    ! [A_40: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(uminus473333897omplex,A_40)),A_40) = zero_zero_complex ).

tff(fact_2181_ab__left__minus,axiom,
    ! [A_40: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,A_40)),A_40) = zero_zero_real ).

tff(fact_2182_ab__left__minus,axiom,
    ! [A_40: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,A_40)),A_40) = zero_zero_int ).

tff(fact_2183_add__eq__0__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Y_2) = zero_zero_rat )
    <=> ( Y_2 = hAPP_rat_rat(uminus_uminus_rat,X_1) ) ) ).

tff(fact_2184_add__eq__0__iff,axiom,
    ! [X_1: complex,Y_2: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Y_2) = zero_zero_complex )
    <=> ( Y_2 = hAPP_complex_complex(uminus473333897omplex,X_1) ) ) ).

tff(fact_2185_add__eq__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real )
    <=> ( Y_2 = hAPP_real_real(uminus_uminus_real,X_1) ) ) ).

tff(fact_2186_add__eq__0__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2) = zero_zero_int )
    <=> ( Y_2 = hAPP_int_int(uminus_uminus_int,X_1) ) ) ).

tff(fact_2187_minus__unique,axiom,
    ! [A_39: rat,B_32: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_39),B_32) = zero_zero_rat )
     => ( hAPP_rat_rat(uminus_uminus_rat,A_39) = B_32 ) ) ).

tff(fact_2188_minus__unique,axiom,
    ! [A_39: complex,B_32: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_39),B_32) = zero_zero_complex )
     => ( hAPP_complex_complex(uminus473333897omplex,A_39) = B_32 ) ) ).

tff(fact_2189_minus__unique,axiom,
    ! [A_39: real,B_32: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_39),B_32) = zero_zero_real )
     => ( hAPP_real_real(uminus_uminus_real,A_39) = B_32 ) ) ).

tff(fact_2190_minus__unique,axiom,
    ! [A_39: int,B_32: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_39),B_32) = zero_zero_int )
     => ( hAPP_int_int(uminus_uminus_int,A_39) = B_32 ) ) ).

tff(fact_2191_diff__0,axiom,
    ! [A_38: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A_38) = hAPP_rat_rat(uminus_uminus_rat,A_38) ).

tff(fact_2192_diff__0,axiom,
    ! [A_38: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,zero_zero_complex),A_38) = hAPP_complex_complex(uminus473333897omplex,A_38) ).

tff(fact_2193_diff__0,axiom,
    ! [A_38: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,zero_zero_real),A_38) = hAPP_real_real(uminus_uminus_real,A_38) ).

tff(fact_2194_diff__0,axiom,
    ! [A_38: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,zero_zero_int),A_38) = hAPP_int_int(uminus_uminus_int,A_38) ).

tff(fact_2195_square__eq__1__iff,axiom,
    ! [X_1: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1) = one_one_rat )
    <=> ( ( X_1 = one_one_rat )
        | ( X_1 = hAPP_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

tff(fact_2196_square__eq__1__iff,axiom,
    ! [X_1: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),X_1) = one_one_complex )
    <=> ( ( X_1 = one_one_complex )
        | ( X_1 = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ) ) ) ).

tff(fact_2197_square__eq__1__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1) = one_one_real )
    <=> ( ( X_1 = one_one_real )
        | ( X_1 = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

tff(fact_2198_square__eq__1__iff,axiom,
    ! [X_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1) = one_one_int )
    <=> ( ( X_1 = one_one_int )
        | ( X_1 = hAPP_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

tff(fact_2199_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
    ! [X_10: rat] : hAPP_rat_rat(uminus_uminus_rat,X_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(uminus_uminus_rat,one_one_rat)),X_10) ).

tff(fact_2200_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
    ! [X_10: complex] : hAPP_complex_complex(uminus473333897omplex,X_10) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(uminus473333897omplex,one_one_complex)),X_10) ).

tff(fact_2201_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
    ! [X_10: real] : hAPP_real_real(uminus_uminus_real,X_10) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,one_one_real)),X_10) ).

tff(fact_2202_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
    ! [X_10: int] : hAPP_int_int(uminus_uminus_int,X_10) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,one_one_int)),X_10) ).

tff(fact_2203_diff__minus__eq__add,axiom,
    ! [A_37: rat,B_31: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_37),hAPP_rat_rat(uminus_uminus_rat,B_31)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_37),B_31) ).

tff(fact_2204_diff__minus__eq__add,axiom,
    ! [A_37: complex,B_31: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_37),hAPP_complex_complex(uminus473333897omplex,B_31)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_37),B_31) ).

tff(fact_2205_diff__minus__eq__add,axiom,
    ! [A_37: real,B_31: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_37),hAPP_real_real(uminus_uminus_real,B_31)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_37),B_31) ).

tff(fact_2206_diff__minus__eq__add,axiom,
    ! [A_37: int,B_31: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_37),hAPP_int_int(uminus_uminus_int,B_31)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_37),B_31) ).

tff(fact_2207_ab__diff__minus,axiom,
    ! [A_36: rat,B_30: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_36),B_30) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_36),hAPP_rat_rat(uminus_uminus_rat,B_30)) ).

tff(fact_2208_ab__diff__minus,axiom,
    ! [A_36: complex,B_30: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_36),B_30) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_36),hAPP_complex_complex(uminus473333897omplex,B_30)) ).

tff(fact_2209_ab__diff__minus,axiom,
    ! [A_36: real,B_30: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_36),B_30) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_36),hAPP_real_real(uminus_uminus_real,B_30)) ).

tff(fact_2210_ab__diff__minus,axiom,
    ! [A_36: int,B_30: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_36),B_30) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_36),hAPP_int_int(uminus_uminus_int,B_30)) ).

tff(fact_2211_Groups_Odiff__def,axiom,
    ! [A_35: rat,B_29: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_35),B_29) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_35),hAPP_rat_rat(uminus_uminus_rat,B_29)) ).

tff(fact_2212_Groups_Odiff__def,axiom,
    ! [A_35: complex,B_29: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_35),B_29) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_35),hAPP_complex_complex(uminus473333897omplex,B_29)) ).

tff(fact_2213_Groups_Odiff__def,axiom,
    ! [A_35: real,B_29: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_35),B_29) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_35),hAPP_real_real(uminus_uminus_real,B_29)) ).

tff(fact_2214_Groups_Odiff__def,axiom,
    ! [A_35: int,B_29: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_35),B_29) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_35),hAPP_int_int(uminus_uminus_int,B_29)) ).

tff(fact_2215_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
    ! [X_9: rat,Y_10: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_9),Y_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_9),hAPP_rat_rat(uminus_uminus_rat,Y_10)) ).

tff(fact_2216_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
    ! [X_9: complex,Y_10: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_9),Y_10) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_9),hAPP_complex_complex(uminus473333897omplex,Y_10)) ).

tff(fact_2217_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
    ! [X_9: real,Y_10: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_9),Y_10) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_9),hAPP_real_real(uminus_uminus_real,Y_10)) ).

tff(fact_2218_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
    ! [X_9: int,Y_10: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_9),Y_10) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_9),hAPP_int_int(uminus_uminus_int,Y_10)) ).

tff(fact_2219_pos__mod__bound,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_2),B_1)),B_1)) ) ).

tff(fact_2220_neg__mod__bound,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),hAPP_int_int(div_mod_int(A_2),B_1))) ) ).

tff(fact_2221_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X),Y))) ) ) ).

tff(fact_2222_zmod__le__nonneg__dividend,axiom,
    ! [K_2: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M_1),K_2)),M_1)) ) ).

tff(fact_2223_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( hAPP_int_int(div_mod_int(M),D) = zero_zero_int )
    <=> ? [Q_3: int] : M = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Q_3) ) ).

tff(fact_2224_abs__zmult__eq__1,axiom,
    ! [M_1: int,N_1: int] :
      ( ( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)) = one_one_int )
     => ( hAPP_int_int(abs_abs_int,M_1) = one_one_int ) ) ).

tff(fact_2225_zadd__zminus__inverse2,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),Z_1) = zero_zero_int ).

tff(fact_2226_bin__less__0__simps_I2_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).

tff(fact_2227_diff__int__def__symmetric,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) ).

tff(fact_2228_diff__int__def,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) ).

tff(fact_2229_zmod__minus1,axiom,
    ! [B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,min)),B_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),one_one_int) ) ) ).

tff(fact_2230_abs__eq__mult,axiom,
    ! [B_28: rat,A_34: rat] :
      ( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_34))
          | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_34),zero_zero_rat)) )
        & ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_28))
          | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_28),zero_zero_rat)) ) )
     => ( hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_34),B_28)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_34)),hAPP_rat_rat(abs_abs_rat,B_28)) ) ) ).

tff(fact_2231_abs__eq__mult,axiom,
    ! [B_28: real,A_34: real] :
      ( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_34))
          | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_34),zero_zero_real)) )
        & ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_28))
          | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_28),zero_zero_real)) ) )
     => ( hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_34),B_28)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_34)),hAPP_real_real(abs_abs_real,B_28)) ) ) ).

tff(fact_2232_abs__eq__mult,axiom,
    ! [B_28: int,A_34: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_34))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_34),zero_zero_int)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_28))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_28),zero_zero_int)) ) )
     => ( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_34),B_28)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_34)),hAPP_int_int(abs_abs_int,B_28)) ) ) ).

tff(fact_2233_abs__mult__pos,axiom,
    ! [Y_9: rat,X_8: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_8))
     => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,Y_9)),X_8) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_9),X_8)) ) ) ).

tff(fact_2234_abs__mult__pos,axiom,
    ! [Y_9: real,X_8: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_8))
     => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,Y_9)),X_8) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_9),X_8)) ) ) ).

tff(fact_2235_abs__mult__pos,axiom,
    ! [Y_9: int,X_8: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_8))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,Y_9)),X_8) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_9),X_8)) ) ) ).

tff(fact_2236_abs__diff__triangle__ineq,axiom,
    ! [A_33: rat,B_27: rat,C_15: rat,D_5: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_33),B_27)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_15),D_5)))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_33),C_15))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_27),D_5))))) ).

tff(fact_2237_abs__diff__triangle__ineq,axiom,
    ! [A_33: real,B_27: real,C_15: real,D_5: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_33),B_27)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_15),D_5)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_33),C_15))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_27),D_5))))) ).

tff(fact_2238_abs__diff__triangle__ineq,axiom,
    ! [A_33: int,B_27: int,C_15: int,D_5: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_33),B_27)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_15),D_5)))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_33),C_15))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_27),D_5))))) ).

tff(fact_2239_abs__triangle__ineq4,axiom,
    ! [A_32: rat,B_26: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_32),B_26))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_32)),hAPP_rat_rat(abs_abs_rat,B_26)))) ).

tff(fact_2240_abs__triangle__ineq4,axiom,
    ! [A_32: real,B_26: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_32),B_26))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_32)),hAPP_real_real(abs_abs_real,B_26)))) ).

tff(fact_2241_abs__triangle__ineq4,axiom,
    ! [A_32: int,B_26: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_32),B_26))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_32)),hAPP_int_int(abs_abs_int,B_26)))) ).

tff(fact_2242_Ints__double__eq__0__iff,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Aa) = zero_zero_real )
      <=> ( Aa = zero_zero_real ) ) ) ).

tff(fact_2243_Ints__double__eq__0__iff,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa) = zero_zero_int )
      <=> ( Aa = zero_zero_int ) ) ) ).

tff(fact_2244_Ints__double__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Aa),Aa) = zero_zero_rat )
      <=> ( Aa = zero_zero_rat ) ) ) ).

tff(fact_2245_Ints__double__eq__0__iff,axiom,
    ! [Aa: complex] :
      ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Aa),ring_1_Ints_complex))
     => ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Aa),Aa) = zero_zero_complex )
      <=> ( Aa = zero_zero_complex ) ) ) ).

tff(fact_2246_zabs__less__one__iff,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_2)),one_one_int))
    <=> ( Z_2 = zero_zero_int ) ) ).

tff(fact_2247_mod__neg__neg__trivial,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_2))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = A_2 ) ) ) ).

tff(fact_2248_neg__mod__conj,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_2),B_1)),zero_zero_int))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),hAPP_int_int(div_mod_int(A_2),B_1))) ) ) ).

tff(fact_2249_neg__mod__sign,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_2),B_1)),zero_zero_int)) ) ).

tff(fact_2250_mod__pos__pos__trivial,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = A_2 ) ) ) ).

tff(fact_2251_pos__mod__conj,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_2),B_1)))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_2),B_1)),B_1)) ) ) ).

tff(fact_2252_pos__mod__sign,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_2),B_1))) ) ).

tff(fact_2253_posDivAlg__0,axiom,
    ! [B_1: int] : posDivAlg(zero_zero_int,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ).

tff(fact_2254_diff__number__of__eq,axiom,
    ! [V_5: int,W_4: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_5)),hAPP_int_rat(number_number_of_rat,W_4)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),hAPP_int_int(uminus_uminus_int,W_4))) ).

tff(fact_2255_diff__number__of__eq,axiom,
    ! [V_5: int,W_4: int] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_5)),hAPP_int_complex(number528085621omplex,W_4)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),hAPP_int_int(uminus_uminus_int,W_4))) ).

tff(fact_2256_diff__number__of__eq,axiom,
    ! [V_5: int,W_4: int] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_5)),hAPP_int_real(number267125858f_real,W_4)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),hAPP_int_int(uminus_uminus_int,W_4))) ).

tff(fact_2257_diff__number__of__eq,axiom,
    ! [V_5: int,W_4: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_5)),hAPP_int_int(number_number_of_int,W_4)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),hAPP_int_int(uminus_uminus_int,W_4))) ).

tff(fact_2258_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M_1: int,N_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1) = one_one_int )
     => ( ( M_1 = one_one_int )
        | ( M_1 = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_2259_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = hAPP_int_int(number_number_of_int,min) )
          & ( N = hAPP_int_int(number_number_of_int,min) ) ) ) ) ).

tff(fact_2260_minus__numeral__code_I6_J,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),hAPP_int_int(uminus_uminus_int,W))) ).

tff(fact_2261_divmod__int__rel__mod,axiom,
    ! [A_2: int,B_1: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( ( B_1 != zero_zero_int )
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = R_3 ) ) ) ).

tff(fact_2262_d22set__g__1,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(d22set,Aa)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ba)) ) ).

tff(fact_2263_mod__pos__neg__trivial,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)),zero_zero_int))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1) ) ) ) ).

tff(fact_2264_add__number__of__diff2,axiom,
    ! [V_4: int,C_14: rat,W_3: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_14),hAPP_int_rat(number_number_of_rat,W_3))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),hAPP_int_int(uminus_uminus_int,W_3)))),C_14) ).

tff(fact_2265_add__number__of__diff2,axiom,
    ! [V_4: int,C_14: complex,W_3: int] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_4)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_14),hAPP_int_complex(number528085621omplex,W_3))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),hAPP_int_int(uminus_uminus_int,W_3)))),C_14) ).

tff(fact_2266_add__number__of__diff2,axiom,
    ! [V_4: int,C_14: real,W_3: int] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_4)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_14),hAPP_int_real(number267125858f_real,W_3))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),hAPP_int_int(uminus_uminus_int,W_3)))),C_14) ).

tff(fact_2267_add__number__of__diff2,axiom,
    ! [V_4: int,C_14: int,W_3: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_4)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_14),hAPP_int_int(number_number_of_int,W_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),hAPP_int_int(uminus_uminus_int,W_3)))),C_14) ).

tff(fact_2268_Ints__odd__nonzero,axiom,
    ! [Aa: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Aa),ring_1_Ints_real))
     => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),Aa)),Aa) != zero_zero_real ) ) ).

tff(fact_2269_Ints__odd__nonzero,axiom,
    ! [Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
     => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Aa)),Aa) != zero_zero_int ) ) ).

tff(fact_2270_Ints__odd__nonzero,axiom,
    ! [Aa: rat] :
      ( hBOOL(hAPP_f311463272l_bool(hAPP_r1889917837l_bool(member_rat,Aa),ring_1_Ints_rat))
     => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),Aa)),Aa) != zero_zero_rat ) ) ).

tff(fact_2271_Ints__odd__nonzero,axiom,
    ! [Aa: complex] :
      ( hBOOL(hAPP_f1734966846l_bool(hAPP_c769186317l_bool(member_complex,Aa),ring_1_Ints_complex))
     => ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),Aa)),Aa) != zero_zero_complex ) ) ).

tff(fact_2272_divmod__int__rel__mod__eq,axiom,
    ! [A_12: int,B_10: int,Q_5: int,Y: int] :
      ( ( A_12 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Q_5)),Y) )
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),B_10)) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),Y))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),zero_zero_int)) ) ) )
       => ( ( B_10 != zero_zero_int )
         => ( hAPP_int_int(div_mod_int(A_12),B_10) = Y ) ) ) ) ).

tff(fact_2273_zmult2__lemma__aux2,axiom,
    ! [Q_1: int,B_1: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(div_mod_int(Q_1),C_1))),R_3)),zero_zero_int)) ) ) ) ).

tff(fact_2274_zmult2__lemma__aux1,axiom,
    ! [Q_1: int,B_1: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(div_mod_int(Q_1),C_1))),R_3))) ) ) ) ).

tff(fact_2275_zmult2__lemma__aux4,axiom,
    ! [Q_1: int,B_1: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(div_mod_int(Q_1),C_1))),R_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C_1))) ) ) ) ).

tff(fact_2276_zmult2__lemma__aux3,axiom,
    ! [Q_1: int,B_1: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(div_mod_int(Q_1),C_1))),R_3))) ) ) ) ).

tff(fact_2277_zmod__eq__0D,axiom,
    ! [M_6: int,D_4: int] :
      ( ( hAPP_int_int(div_mod_int(M_6),D_4) = zero_zero_int )
     => ? [Q_3: int] : M_6 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_3) ) ).

tff(fact_2278_pdivmod__posDivAlg,axiom,
    ! [K_2: int,L_2: int] :
      ( ( ( L_2 = zero_zero_int )
       => ( pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(abs_abs_int,K_2)) ) )
      & ( ( L_2 != zero_zero_int )
       => ( pdivmod(K_2,L_2) = posDivAlg(hAPP_int_int(abs_abs_int,K_2),hAPP_int_int(abs_abs_int,L_2)) ) ) ) ).

tff(fact_2279_z3mod__def,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),L_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) ) ) ) ).

tff(fact_2280_xzgcda_Osimps,axiom,
    ! [M_1: int,N_1: int,R_5: int,S_4: int,S_2: int,T_3: int,T_2: int,R_3: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
       => ( xzgcda(M_1,N_1,R_5,R_3,S_4,S_2,T_3,T_2) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_5),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_4),T_3)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
       => ( xzgcda(M_1,N_1,R_5,R_3,S_4,S_2,T_3,T_2) = xzgcda(M_1,N_1,R_3,hAPP_int_int(div_mod_int(R_5),R_3),S_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_2)),T_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_2))) ) ) ) ).

tff(fact_2281_nat0__intermed__int__val,axiom,
    ! [K_1: int,F: fun_nat_int,N: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N)))
         => ? [I_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
              & ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).

tff(fact_2282_int__val__lemma,axiom,
    ! [K_1: int,F: fun_nat_int,N: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N)))
         => ? [I_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
              & ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).

tff(fact_2283_nat__intermed__int__val,axiom,
    ! [K_1: int,F: fun_nat_int,N: nat,M: nat] :
      ( ! [I_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),I_2))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N)) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,M)),K_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N)))
           => ? [I_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),I_2))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
                & ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ) ).

tff(fact_2284_negDivAlg__minus1,axiom,
    ! [B_1: int] : negDivAlg(hAPP_int_int(number_number_of_int,min),B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),one_one_int)) ).

tff(fact_2285_zmult2__lemma,axiom,
    ! [C_1: int,A_2: int,B_1: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( ( B_1 != zero_zero_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
         => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C_1)),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Q_1),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(div_mod_int(Q_1),C_1))),R_3)))) ) ) ) ).

tff(fact_2286_not__real__square__gt__zero,axiom,
    ! [X_1: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)))
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_2287_mod__le__divisor,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M_1),N_1)),N_1)) ) ).

tff(fact_2288_mod__eq__0__iff,axiom,
    ! [M: nat,D: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M),D) = zero_zero_nat )
    <=> ? [Q_3: nat] : M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D),Q_3) ) ).

tff(fact_2289_mult__eq__if,axiom,
    ! [N_1: nat,M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = zero_zero_nat ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1)) ) ) ) ).

tff(fact_2290_mod__if,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
       => ( hAPP_nat_nat(div_mod_nat(M_1),N_1) = M_1 ) )
      & ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
       => ( hAPP_nat_nat(div_mod_nat(M_1),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1) ) ) ) ).

tff(fact_2291_mod__geq,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( hAPP_nat_nat(div_mod_nat(M_1),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1) ) ) ).

tff(fact_2292_mod__less__divisor,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M_1),N_1)),N_1)) ) ).

tff(fact_2293_mod__less,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( hAPP_nat_nat(div_mod_nat(M_1),N_1) = M_1 ) ) ).

tff(fact_2294_mod__lemma,axiom,
    ! [Q_1: nat,R_3: nat,B_1: nat,C_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),B_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),hAPP_nat_nat(div_mod_nat(Q_1),C_1))),R_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C_1))) ) ) ).

tff(fact_2295_split__mod,axiom,
    ! [P_1: fun_nat_bool,N: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(N),K_1)))
    <=> ( ( ( K_1 = zero_zero_nat )
         => hBOOL(hAPP_nat_bool(P_1,N)) )
        & ( ( K_1 != zero_zero_nat )
         => ! [I_2: nat,J_1: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
             => ( ( N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) )
               => hBOOL(hAPP_nat_bool(P_1,J_1)) ) ) ) ) ) ).

tff(fact_2296_real__abs__def,axiom,
    ! [R_3: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,R_3) = hAPP_real_real(uminus_uminus_real,R_3) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,R_3) = R_3 ) ) ) ).

tff(fact_2297_abs__real__def,axiom,
    ! [A_2: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,A_2) = hAPP_real_real(uminus_uminus_real,A_2) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,A_2) = A_2 ) ) ) ).

tff(fact_2298_mod__mult__self3,axiom,
    ! [K_2: nat,N_1: nat,M_1: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)),M_1)),N_1) = hAPP_nat_nat(div_mod_nat(M_1),N_1) ).

tff(fact_2299_le__mod__geq,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_nat_nat(div_mod_nat(M_1),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1) ) ) ).

tff(fact_2300_mod__less__eq__dividend,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M_1),N_1)),M_1)) ).

tff(fact_2301_abs__sum__triangle__ineq,axiom,
    ! [X: real,Y: real,L_2: real,M_1: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,L_2)),hAPP_real_real(uminus_uminus_real,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,L_2)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,M_1)))))) ).

tff(fact_2302_abs__le__interval__iff,axiom,
    ! [X_1: real,R_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),R_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,R_2)),X_1))
        & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),R_2)) ) ) ).

tff(fact_2303_termination__basic__simps_I3_J,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_2304_termination__basic__simps_I4_J,axiom,
    ! [Y: nat,X: nat,Z_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_2305_real__diff__def,axiom,
    ! [R_3: real,S_2: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),S_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),hAPP_real_real(uminus_uminus_real,S_2)) ).

tff(fact_2306_minus__real__def,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y)) ).

tff(fact_2307_abs__minus__add__cancel,axiom,
    ! [X: real,Y: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y))) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,X))) ).

tff(fact_2308_abs__add__one__not__less__self,axiom,
    ! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,X)),one_one_real)),X)) ).

tff(fact_2309_termination__basic__simps_I5_J,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).

tff(fact_2310_termination__basic__simps_I2_J,axiom,
    ! [Y: nat,X: nat,Z_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_2311_termination__basic__simps_I1_J,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_2312_zdiv__zero,axiom,
    ! [B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),B_1) = zero_zero_int ).

tff(fact_2313_abs__add__one__gt__zero,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(abs_abs_real,X)))) ).

tff(fact_2314_real__0__less__add__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,X_1)),Y_2)) ) ).

tff(fact_2315_real__add__less__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(uminus_uminus_real,X_1))) ) ).

tff(fact_2316_real__add__eq__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real )
    <=> ( Y_2 = hAPP_real_real(uminus_uminus_real,X_1) ) ) ).

tff(fact_2317_real__add__minus__iff,axiom,
    ! [X_1: real,Aa: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,Aa)) = zero_zero_real )
    <=> ( X_1 = Aa ) ) ).

tff(fact_2318_zdiv__zminus2,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_2)),B_1) ).

tff(fact_2319_zdiv__zminus__zminus,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) ).

tff(fact_2320_div__by__0,axiom,
    ! [A_31: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_31),zero_z891286103de_int) = zero_z891286103de_int ).

tff(fact_2321_div__by__0,axiom,
    ! [A_31: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_31),zero_z126310315umeral) = zero_z126310315umeral ).

tff(fact_2322_div__by__0,axiom,
    ! [A_31: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_31),zero_zero_nat) = zero_zero_nat ).

tff(fact_2323_div__by__0,axiom,
    ! [A_31: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_31),zero_zero_int) = zero_zero_int ).

tff(fact_2324_div__0,axiom,
    ! [A_30: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,zero_z891286103de_int),A_30) = zero_z891286103de_int ).

tff(fact_2325_div__0,axiom,
    ! [A_30: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,zero_z126310315umeral),A_30) = zero_z126310315umeral ).

tff(fact_2326_div__0,axiom,
    ! [A_30: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,zero_zero_nat),A_30) = zero_zero_nat ).

tff(fact_2327_div__0,axiom,
    ! [A_30: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),A_30) = zero_zero_int ).

tff(fact_2328_div__by__1,axiom,
    ! [A_29: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_29),one_on1684967323de_int) = A_29 ).

tff(fact_2329_div__by__1,axiom,
    ! [A_29: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_29),one_on1645066479umeral) = A_29 ).

tff(fact_2330_div__by__1,axiom,
    ! [A_29: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_29),one_one_nat) = A_29 ).

tff(fact_2331_div__by__1,axiom,
    ! [A_29: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_29),one_one_int) = A_29 ).

tff(fact_2332_div__neg__pos__less0,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),zero_zero_int)) ) ) ).

tff(fact_2333_neg__imp__zdiv__neg__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Aa),Ba)),zero_zero_int))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ) ).

tff(fact_2334_pos__imp__zdiv__neg__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Aa),Ba)),zero_zero_int))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ) ).

tff(fact_2335_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))) ) ) ).

tff(fact_2336_zdiv__self,axiom,
    ! [A_2: int] :
      ( ( A_2 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),A_2) = one_one_int ) ) ).

tff(fact_2337_zmod__zdiv__trivial,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_2),B_1)),B_1) = zero_zero_int ).

tff(fact_2338_DIVISION__BY__ZERO,axiom,
    ! [A_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),zero_zero_int) = zero_zero_int )
      & ( hAPP_int_int(div_mod_int(A_2),zero_zero_int) = A_2 ) ) ).

tff(fact_2339_real__add__le__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),hAPP_real_real(uminus_uminus_real,X_1))) ) ).

tff(fact_2340_real__0__le__add__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,X_1)),Y_2)) ) ).

tff(fact_2341_real__minus__mult__self__le,axiom,
    ! [U_1: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),U_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X))) ).

tff(fact_2342_zdiv__zadd1__eq,axiom,
    ! [A_2: int,B_1: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_2),C_1)),hAPP_int_int(div_mod_int(B_1),C_1))),C_1)) ).

tff(fact_2343_pdivmod__def,axiom,
    ! [K_2: int,L_2: int] : pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))) ).

tff(fact_2344_div__mult__mult1__if,axiom,
    ! [A_28: quickcheck_code_int,B_25: quickcheck_code_int,C_13: quickcheck_code_int] :
      ( ( ( C_13 = zero_z891286103de_int )
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_13),A_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_13),B_25)) = zero_z891286103de_int ) )
      & ( ( C_13 != zero_z891286103de_int )
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_13),A_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_13),B_25)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_28),B_25) ) ) ) ).

tff(fact_2345_div__mult__mult1__if,axiom,
    ! [A_28: code_code_numeral,B_25: code_code_numeral,C_13: code_code_numeral] :
      ( ( ( C_13 = zero_z126310315umeral )
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_13),A_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_13),B_25)) = zero_z126310315umeral ) )
      & ( ( C_13 != zero_z126310315umeral )
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_13),A_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_13),B_25)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_28),B_25) ) ) ) ).

tff(fact_2346_div__mult__mult1__if,axiom,
    ! [A_28: nat,B_25: nat,C_13: nat] :
      ( ( ( C_13 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_13),A_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_13),B_25)) = zero_zero_nat ) )
      & ( ( C_13 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_13),A_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_13),B_25)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_28),B_25) ) ) ) ).

tff(fact_2347_div__mult__mult1__if,axiom,
    ! [A_28: int,B_25: int,C_13: int] :
      ( ( ( C_13 = zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_13),A_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_13),B_25)) = zero_zero_int ) )
      & ( ( C_13 != zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_13),A_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_13),B_25)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_28),B_25) ) ) ) ).

tff(fact_2348_div__mult__self2__is__id,axiom,
    ! [A_27: quickcheck_code_int,B_24: quickcheck_code_int] :
      ( ( B_24 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_27),B_24)),B_24) = A_27 ) ) ).

tff(fact_2349_div__mult__self2__is__id,axiom,
    ! [A_27: code_code_numeral,B_24: code_code_numeral] :
      ( ( B_24 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_27),B_24)),B_24) = A_27 ) ) ).

tff(fact_2350_div__mult__self2__is__id,axiom,
    ! [A_27: nat,B_24: nat] :
      ( ( B_24 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),B_24)),B_24) = A_27 ) ) ).

tff(fact_2351_div__mult__self2__is__id,axiom,
    ! [A_27: int,B_24: int] :
      ( ( B_24 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),B_24)),B_24) = A_27 ) ) ).

tff(fact_2352_div__mult__self1__is__id,axiom,
    ! [A_26: quickcheck_code_int,B_23: quickcheck_code_int] :
      ( ( B_23 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_23),A_26)),B_23) = A_26 ) ) ).

tff(fact_2353_div__mult__self1__is__id,axiom,
    ! [A_26: code_code_numeral,B_23: code_code_numeral] :
      ( ( B_23 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_23),A_26)),B_23) = A_26 ) ) ).

tff(fact_2354_div__mult__self1__is__id,axiom,
    ! [A_26: nat,B_23: nat] :
      ( ( B_23 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_23),A_26)),B_23) = A_26 ) ) ).

tff(fact_2355_div__mult__self1__is__id,axiom,
    ! [A_26: int,B_23: int] :
      ( ( B_23 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_23),A_26)),B_23) = A_26 ) ) ).

tff(fact_2356_div__mult__mult2,axiom,
    ! [A_25: quickcheck_code_int,B_22: quickcheck_code_int,C_12: quickcheck_code_int] :
      ( ( C_12 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_25),C_12)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_22),C_12)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_25),B_22) ) ) ).

tff(fact_2357_div__mult__mult2,axiom,
    ! [A_25: code_code_numeral,B_22: code_code_numeral,C_12: code_code_numeral] :
      ( ( C_12 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_25),C_12)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_22),C_12)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_25),B_22) ) ) ).

tff(fact_2358_div__mult__mult2,axiom,
    ! [A_25: nat,B_22: nat,C_12: nat] :
      ( ( C_12 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_25),C_12)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_22),C_12)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_25),B_22) ) ) ).

tff(fact_2359_div__mult__mult2,axiom,
    ! [A_25: int,B_22: int,C_12: int] :
      ( ( C_12 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_25),C_12)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_22),C_12)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_25),B_22) ) ) ).

tff(fact_2360_div__mult__mult1,axiom,
    ! [A_24: quickcheck_code_int,B_21: quickcheck_code_int,C_11: quickcheck_code_int] :
      ( ( C_11 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_11),A_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_11),B_21)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_24),B_21) ) ) ).

tff(fact_2361_div__mult__mult1,axiom,
    ! [A_24: code_code_numeral,B_21: code_code_numeral,C_11: code_code_numeral] :
      ( ( C_11 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_11),A_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_11),B_21)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_24),B_21) ) ) ).

tff(fact_2362_div__mult__mult1,axiom,
    ! [A_24: nat,B_21: nat,C_11: nat] :
      ( ( C_11 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_11),A_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_11),B_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_24),B_21) ) ) ).

tff(fact_2363_div__mult__mult1,axiom,
    ! [A_24: int,B_21: int,C_11: int] :
      ( ( C_11 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_11),A_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_11),B_21)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_24),B_21) ) ) ).

tff(fact_2364_div__self,axiom,
    ! [A_23: quickcheck_code_int] :
      ( ( A_23 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_23),A_23) = one_on1684967323de_int ) ) ).

tff(fact_2365_div__self,axiom,
    ! [A_23: code_code_numeral] :
      ( ( A_23 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_23),A_23) = one_on1645066479umeral ) ) ).

tff(fact_2366_div__self,axiom,
    ! [A_23: nat] :
      ( ( A_23 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_23),A_23) = one_one_nat ) ) ).

tff(fact_2367_div__self,axiom,
    ! [A_23: int] :
      ( ( A_23 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_23),A_23) = one_one_int ) ) ).

tff(fact_2368_mod__div__trivial,axiom,
    ! [A_22: quickcheck_code_int,B_20: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_22),B_20)),B_20) = zero_z891286103de_int ).

tff(fact_2369_mod__div__trivial,axiom,
    ! [A_22: code_code_numeral,B_20: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_22),B_20)),B_20) = zero_z126310315umeral ).

tff(fact_2370_mod__div__trivial,axiom,
    ! [A_22: nat,B_20: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(div_mod_nat(A_22),B_20)),B_20) = zero_zero_nat ).

tff(fact_2371_mod__div__trivial,axiom,
    ! [A_22: int,B_20: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_22),B_20)),B_20) = zero_zero_int ).

tff(fact_2372_negDivAlg__div__mod,axiom,
    ! [L_2: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L_2))
       => ( negDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) ) ) ) ).

tff(fact_2373_zdiv__eq__0__iff,axiom,
    ! [I_1: int,K_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1) = zero_zero_int )
    <=> ( ( K_1 = zero_zero_int )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I_1))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) ) ) ) ).

tff(fact_2374_pos__imp__zdiv__nonneg__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Aa),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ) ).

tff(fact_2375_pos__imp__zdiv__pos__iff,axiom,
    ! [I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1)) ) ) ).

tff(fact_2376_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Aa),Ba)))
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba)) ) ) ) ).

tff(fact_2377_zdiv__mono2,axiom,
    ! [B_1: int,B_4: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_4))) ) ) ) ).

tff(fact_2378_div__nonneg__neg__le0,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),zero_zero_int)) ) ) ).

tff(fact_2379_div__pos__pos__trivial,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = zero_zero_int ) ) ) ).

tff(fact_2380_neg__imp__zdiv__nonneg__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Aa),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ) ).

tff(fact_2381_div__nonpos__pos__le0,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),zero_zero_int)) ) ) ).

tff(fact_2382_zdiv__mono2__neg,axiom,
    ! [B_1: int,B_4: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_4)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))) ) ) ) ).

tff(fact_2383_div__neg__neg__trivial,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = zero_zero_int ) ) ) ).

tff(fact_2384_zdiv__mono1,axiom,
    ! [B_1: int,A_2: int,A_4: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),A_4))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_4),B_1))) ) ) ).

tff(fact_2385_zdiv__mono1__neg,axiom,
    ! [B_1: int,A_2: int,A_4: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),A_4))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_4),B_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))) ) ) ).

tff(fact_2386_int__div__less__self,axiom,
    ! [K_2: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),K_2)),X)) ) ) ).

tff(fact_2387_zdiv__zmult2__eq,axiom,
    ! [A_2: int,B_1: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),C_1) ) ) ).

tff(fact_2388_div__prop1,axiom,
    ! [X: int,Y: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y)) ) ) ).

tff(fact_2389_zmod__zdiv__equality,axiom,
    ! [A_2: int,B_1: int] : A_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))),hAPP_int_int(div_mod_int(A_2),B_1)) ).

tff(fact_2390_zdiv__zmult1__eq,axiom,
    ! [A_2: int,B_1: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(div_mod_int(B_1),C_1))),C_1)) ).

tff(fact_2391_zdiv__zmod__equality,axiom,
    ! [B_1: int,A_2: int,K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))),hAPP_int_int(div_mod_int(A_2),B_1))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),K_2) ).

tff(fact_2392_zdiv__zmod__equality2,axiom,
    ! [A_2: int,B_1: int,K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),B_1)),hAPP_int_int(div_mod_int(A_2),B_1))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),K_2) ).

tff(fact_2393_zmod__zdiv__equality_H,axiom,
    ! [M_1: int,N_1: int] : hAPP_int_int(div_mod_int(M_1),N_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M_1),N_1)),N_1)) ).

tff(fact_2394_zmult__div__cancel,axiom,
    ! [N_1: int,M_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M_1),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M_1),hAPP_int_int(div_mod_int(M_1),N_1)) ).

tff(fact_2395_zdiv__minus1__right,axiom,
    ! [A_2: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(uminus_uminus_int,A_2) ).

tff(fact_2396_div__mult__self2,axiom,
    ! [A_21: quickcheck_code_int,C_10: quickcheck_code_int,B_19: quickcheck_code_int] :
      ( ( B_19 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_21),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_19),C_10))),B_19) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_10),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_21),B_19)) ) ) ).

tff(fact_2397_div__mult__self2,axiom,
    ! [A_21: code_code_numeral,C_10: code_code_numeral,B_19: code_code_numeral] :
      ( ( B_19 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_21),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_19),C_10))),B_19) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_10),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_21),B_19)) ) ) ).

tff(fact_2398_div__mult__self2,axiom,
    ! [A_21: nat,C_10: nat,B_19: nat] :
      ( ( B_19 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_19),C_10))),B_19) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_10),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_21),B_19)) ) ) ).

tff(fact_2399_div__mult__self2,axiom,
    ! [A_21: int,C_10: int,B_19: int] :
      ( ( B_19 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_21),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_19),C_10))),B_19) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_10),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_21),B_19)) ) ) ).

tff(fact_2400_div__mult__self1,axiom,
    ! [A_20: quickcheck_code_int,C_9: quickcheck_code_int,B_18: quickcheck_code_int] :
      ( ( B_18 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_20),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_9),B_18))),B_18) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_9),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_20),B_18)) ) ) ).

tff(fact_2401_div__mult__self1,axiom,
    ! [A_20: code_code_numeral,C_9: code_code_numeral,B_18: code_code_numeral] :
      ( ( B_18 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_20),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_9),B_18))),B_18) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_9),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_20),B_18)) ) ) ).

tff(fact_2402_div__mult__self1,axiom,
    ! [A_20: nat,C_9: nat,B_18: nat] :
      ( ( B_18 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_20),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_9),B_18))),B_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_20),B_18)) ) ) ).

tff(fact_2403_div__mult__self1,axiom,
    ! [A_20: int,C_9: int,B_18: int] :
      ( ( B_18 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_20),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_9),B_18))),B_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_9),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_20),B_18)) ) ) ).

tff(fact_2404_divmod__int__rel__div,axiom,
    ! [A_2: int,B_1: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( ( B_1 != zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = Q_1 ) ) ) ).

tff(fact_2405_div__add__self2,axiom,
    ! [A_19: quickcheck_code_int,B_17: quickcheck_code_int] :
      ( ( B_17 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_19),B_17)),B_17) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_19),B_17)),one_on1684967323de_int) ) ) ).

tff(fact_2406_div__add__self2,axiom,
    ! [A_19: code_code_numeral,B_17: code_code_numeral] :
      ( ( B_17 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_19),B_17)),B_17) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_19),B_17)),one_on1645066479umeral) ) ) ).

tff(fact_2407_div__add__self2,axiom,
    ! [A_19: nat,B_17: nat] :
      ( ( B_17 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_19),B_17)),B_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_19),B_17)),one_one_nat) ) ) ).

tff(fact_2408_div__add__self2,axiom,
    ! [A_19: int,B_17: int] :
      ( ( B_17 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_19),B_17)),B_17) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_19),B_17)),one_one_int) ) ) ).

tff(fact_2409_div__add__self1,axiom,
    ! [A_18: quickcheck_code_int,B_16: quickcheck_code_int] :
      ( ( B_16 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_16),A_18)),B_16) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_18),B_16)),one_on1684967323de_int) ) ) ).

tff(fact_2410_div__add__self1,axiom,
    ! [A_18: code_code_numeral,B_16: code_code_numeral] :
      ( ( B_16 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_16),A_18)),B_16) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_18),B_16)),one_on1645066479umeral) ) ) ).

tff(fact_2411_div__add__self1,axiom,
    ! [A_18: nat,B_16: nat] :
      ( ( B_16 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_16),A_18)),B_16) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_18),B_16)),one_one_nat) ) ) ).

tff(fact_2412_div__add__self1,axiom,
    ! [A_18: int,B_16: int] :
      ( ( B_16 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_16),A_18)),B_16) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_18),B_16)),one_one_int) ) ) ).

tff(fact_2413_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_17: quickcheck_code_int,B_15: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_17),B_15)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_17),B_15)),B_15)) = A_17 ).

tff(fact_2414_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_17: code_code_numeral,B_15: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_17),B_15)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_17),B_15)),B_15)) = A_17 ).

tff(fact_2415_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_17: nat,B_15: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_17),B_15)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_17),B_15)),B_15)) = A_17 ).

tff(fact_2416_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_17: int,B_15: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_17),B_15)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_17),B_15)),B_15)) = A_17 ).

tff(fact_2417_mod__div__equality2,axiom,
    ! [B_14: quickcheck_code_int,A_16: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_14),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_16),B_14))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_16),B_14)) = A_16 ).

tff(fact_2418_mod__div__equality2,axiom,
    ! [B_14: code_code_numeral,A_16: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_14),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_16),B_14))),hAPP_C498520661umeral(div_mo1740067990umeral(A_16),B_14)) = A_16 ).

tff(fact_2419_mod__div__equality2,axiom,
    ! [B_14: nat,A_16: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_16),B_14))),hAPP_nat_nat(div_mod_nat(A_16),B_14)) = A_16 ).

tff(fact_2420_mod__div__equality2,axiom,
    ! [B_14: int,A_16: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_14),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_16),B_14))),hAPP_int_int(div_mod_int(A_16),B_14)) = A_16 ).

tff(fact_2421_mod__div__equality,axiom,
    ! [A_15: quickcheck_code_int,B_13: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_15),B_13)),B_13)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_15),B_13)) = A_15 ).

tff(fact_2422_mod__div__equality,axiom,
    ! [A_15: code_code_numeral,B_13: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_15),B_13)),B_13)),hAPP_C498520661umeral(div_mo1740067990umeral(A_15),B_13)) = A_15 ).

tff(fact_2423_mod__div__equality,axiom,
    ! [A_15: nat,B_13: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_15),B_13)),B_13)),hAPP_nat_nat(div_mod_nat(A_15),B_13)) = A_15 ).

tff(fact_2424_mod__div__equality,axiom,
    ! [A_15: int,B_13: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_15),B_13)),B_13)),hAPP_int_int(div_mod_int(A_15),B_13)) = A_15 ).

tff(fact_2425_div__mod__equality2,axiom,
    ! [B_12: quickcheck_code_int,A_14: quickcheck_code_int,C_8: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_12),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_14),B_12))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_14),B_12))),C_8) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_14),C_8) ).

tff(fact_2426_div__mod__equality2,axiom,
    ! [B_12: code_code_numeral,A_14: code_code_numeral,C_8: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_12),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_14),B_12))),hAPP_C498520661umeral(div_mo1740067990umeral(A_14),B_12))),C_8) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_14),C_8) ).

tff(fact_2427_div__mod__equality2,axiom,
    ! [B_12: nat,A_14: nat,C_8: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_12),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_14),B_12))),hAPP_nat_nat(div_mod_nat(A_14),B_12))),C_8) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_14),C_8) ).

tff(fact_2428_div__mod__equality2,axiom,
    ! [B_12: int,A_14: int,C_8: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_12),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_14),B_12))),hAPP_int_int(div_mod_int(A_14),B_12))),C_8) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_14),C_8) ).

tff(fact_2429_div__mod__equality,axiom,
    ! [A_13: quickcheck_code_int,B_11: quickcheck_code_int,C_7: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_13),B_11)),B_11)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_13),B_11))),C_7) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_13),C_7) ).

tff(fact_2430_div__mod__equality,axiom,
    ! [A_13: code_code_numeral,B_11: code_code_numeral,C_7: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_13),B_11)),B_11)),hAPP_C498520661umeral(div_mo1740067990umeral(A_13),B_11))),C_7) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_13),C_7) ).

tff(fact_2431_div__mod__equality,axiom,
    ! [A_13: nat,B_11: nat,C_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_13),B_11)),B_11)),hAPP_nat_nat(div_mod_nat(A_13),B_11))),C_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),C_7) ).

tff(fact_2432_div__mod__equality,axiom,
    ! [A_13: int,B_11: int,C_7: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_13),B_11)),B_11)),hAPP_int_int(div_mod_int(A_13),B_11))),C_7) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),C_7) ).

tff(fact_2433_zdiv__leq__prop,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))),X)) ) ).

tff(fact_2434_div__eq__minus1,axiom,
    ! [B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,min)),B_1) = hAPP_int_int(number_number_of_int,min) ) ) ).

tff(fact_2435_zcong__zmod__aux,axiom,
    ! [A_2: int,B_1: int,M_1: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),M_1)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_2),M_1)),hAPP_int_int(div_mod_int(B_1),M_1))) ).

tff(fact_2436_xzgcda__linear__aux2,axiom,
    ! [R_3: int,S_2: int,T_2: int,R_5: int,S_4: int,M_1: int,T_3: int,N_1: int] :
      ( ( R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_4),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_3),N_1)) )
     => ( ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N_1)) )
       => ( hAPP_int_int(div_mod_int(R_5),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_2))),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_2))),N_1)) ) ) ) ).

tff(fact_2437_divmod__int__rel__div__mod,axiom,
    ! [A_2: int,B_1: int] :
      ( ( B_1 != zero_zero_int )
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),hAPP_int_int(div_mod_int(A_2),B_1)))) ) ).

tff(fact_2438_negDivAlg__correct,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),negDivAlg(A_2,B_1))) ) ) ).

tff(fact_2439_split__zdiv,axiom,
    ! [P_1: fun_int_bool,N: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)))
    <=> ( ( ( K_1 = zero_zero_int )
         => hBOOL(hAPP_int_bool(P_1,zero_zero_int)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
                & ( N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,I_2)) ) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
                & ( N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ) ).

tff(fact_2440_divmod__int__rel__div__eq,axiom,
    ! [A_12: int,B_10: int,Y: int,R_6: int] :
      ( ( A_12 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Y)),R_6) )
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_6))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_6),B_10)) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),R_6))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_6),zero_zero_int)) ) ) )
       => ( ( B_10 != zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_12),B_10) = Y ) ) ) ) ).

tff(fact_2441_div__prop2,axiom,
    ! [X: int,Y: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1)),Z_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y)) ) ) ).

tff(fact_2442_zmod__zmult2__eq,axiom,
    ! [A_2: int,B_1: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hAPP_int_int(div_mod_int(A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),C_1))),hAPP_int_int(div_mod_int(A_2),B_1)) ) ) ).

tff(fact_2443_zdiv__zminus2__eq__if,axiom,
    ! [A_2: int,B_1: int] :
      ( ( B_1 != zero_zero_int )
     => ( ( ( hAPP_int_int(div_mod_int(A_2),B_1) = zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)) ) )
        & ( ( hAPP_int_int(div_mod_int(A_2),B_1) != zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))),one_one_int) ) ) ) ) ).

tff(fact_2444_zdiv__zminus1__eq__if,axiom,
    ! [A_2: int,B_1: int] :
      ( ( B_1 != zero_zero_int )
     => ( ( ( hAPP_int_int(div_mod_int(A_2),B_1) = zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_2)),B_1) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)) ) )
        & ( ( hAPP_int_int(div_mod_int(A_2),B_1) != zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_2)),B_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))),one_one_int) ) ) ) ) ).

tff(fact_2445_zadd1__lemma,axiom,
    ! [B_1: int,Bq: int,Br: int,A_2: int,C_1: int,Aq: int,Ar: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aq),Ar)))
     => ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_1,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Bq),Br)))
       => ( ( C_1 != zero_zero_int )
         => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aq),Bq)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1)))) ) ) ) ).

tff(fact_2446_posDivAlg__div__mod,axiom,
    ! [L_2: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( posDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) ) ) ) ).

tff(fact_2447_div__pos__neg__trivial,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)),zero_zero_int))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_2448_split__pos__lemma,axiom,
    ! [P_1: fun_int_fun_int_bool,N: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)),hAPP_int_int(div_mod_int(N),K_1)))
      <=> ! [I_2: int,J_1: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
              & ( N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ).

tff(fact_2449_split__neg__lemma,axiom,
    ! [P_1: fun_int_fun_int_bool,N: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),K_1)),hAPP_int_int(div_mod_int(N),K_1)))
      <=> ! [I_2: int,J_1: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
              & ( N = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ).

tff(fact_2450_zmult1__lemma,axiom,
    ! [A_2: int,B_1: int,C_1: int,Q_1: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_1,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)))
     => ( ( C_1 != zero_zero_int )
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),Q_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),R_3)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),R_3)),C_1)))) ) ) ).

tff(fact_2451_z3div__def,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),hAPP_int_int(uminus_uminus_int,L_2))) ) ) ) ).

tff(fact_2452_le0,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).

tff(fact_2453_less__zeroE,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).

tff(fact_2454_mod__div__decomp,axiom,
    ! [A_11: quickcheck_code_int,B_9: quickcheck_code_int] : A_11 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_11),B_9)),B_9)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_11),B_9)) ).

tff(fact_2455_mod__div__decomp,axiom,
    ! [A_11: code_code_numeral,B_9: code_code_numeral] : A_11 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_11),B_9)),B_9)),hAPP_C498520661umeral(div_mo1740067990umeral(A_11),B_9)) ).

tff(fact_2456_mod__div__decomp,axiom,
    ! [A_11: nat,B_9: nat] : A_11 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_11),B_9)),B_9)),hAPP_nat_nat(div_mod_nat(A_11),B_9)) ).

tff(fact_2457_mod__div__decomp,axiom,
    ! [A_11: int,B_9: int] : A_11 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_11),B_9)),B_9)),hAPP_int_int(div_mod_int(A_11),B_9)) ).

tff(fact_2458_sin__bound__lemma,axiom,
    ! [U_1: real,V_2: real,X: real,Y: real] :
      ( ( X = Y )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),U_1)),Y))),V_2)) ) ) ).

tff(fact_2459_nat__less__add__iff2,axiom,
    ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N))) ) ) ).

tff(fact_2460_nat__less__add__iff1,axiom,
    ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),M)),N)) ) ) ).

tff(fact_2461_rabs__ratiotest__lemma,axiom,
    ! [X: real,Y: real,C_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y))))
       => ( X = zero_zero_real ) ) ) ).

tff(fact_2462_mult__le__cancel1,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).

tff(fact_2463_nat__mult__div__cancel__disj,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] :
      ( ( ( K_2 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = zero_zero_nat ) )
      & ( ( K_2 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) ) ) ) ).

tff(fact_2464_div__mult2__eq,axiom,
    ! [A_2: nat,B_1: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),B_1)),C_1) ).

tff(fact_2465_mod__mult__distrib2,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M_1),N_1)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).

tff(fact_2466_mod__mult__distrib,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M_1),N_1)),K_2) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).

tff(fact_2467_div__mod__equality_H,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(div_mod_nat(M_1),N_1)) ).

tff(fact_2468_mult__div__cancel,axiom,
    ! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(div_mod_nat(M_1),N_1)) ).

tff(fact_2469_Divides_Omod__div__equality_H,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(div_mod_nat(M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)),N_1)) ).

tff(fact_2470_diff__mult__distrib2,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).

tff(fact_2471_diff__mult__distrib,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).

tff(fact_2472_diff__commute,axiom,
    ! [I: nat,J_2: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K_2)),J_2) ).

tff(fact_2473_nat__mult__commute,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M_1) ).

tff(fact_2474_nat__mult__assoc,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).

tff(fact_2475_div__add1__eq,axiom,
    ! [A_2: nat,B_1: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),B_1)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_1),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_2),C_1)),hAPP_nat_nat(div_mod_nat(B_1),C_1))),C_1)) ).

tff(fact_2476_div__mult1__eq,axiom,
    ! [A_2: nat,B_1: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_1),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),hAPP_nat_nat(div_mod_nat(B_1),C_1))),C_1)) ).

tff(fact_2477_mod__mult2__eq,axiom,
    ! [A_2: nat,B_1: nat,C_1: nat] : hAPP_nat_nat(div_mod_nat(A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),B_1)),C_1))),hAPP_nat_nat(div_mod_nat(A_2),B_1)) ).

tff(fact_2478_div__le__mono,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),K_2))) ) ).

tff(fact_2479_div__le__dividend,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)),M_1)) ).

tff(fact_2480_nat__mult__div__cancel1,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) ) ) ).

tff(fact_2481_div__mult__self__is__m,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),N_1) = M_1 ) ) ).

tff(fact_2482_div__mult__self1__is__m,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M_1)),N_1) = M_1 ) ) ).

tff(fact_2483_div__less,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = zero_zero_nat ) ) ).

tff(fact_2484_split__div,axiom,
    ! [P_1: fun_nat_bool,N: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),K_1)))
    <=> ( ( ( K_1 = zero_zero_nat )
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
        & ( ( K_1 != zero_zero_nat )
         => ! [I_2: nat,J_1: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
             => ( ( N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) )
               => hBOOL(hAPP_nat_bool(P_1,I_2)) ) ) ) ) ) ).

tff(fact_2485_div__le__mono2,axiom,
    ! [K_2: nat,N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),M_1))) ) ) ).

tff(fact_2486_div__less__dividend,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)),M_1)) ) ) ).

tff(fact_2487_mult__0,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N_1) = zero_zero_nat ).

tff(fact_2488_mult__0__right,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),zero_zero_nat) = zero_zero_nat ).

tff(fact_2489_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

tff(fact_2490_nat__mult__eq__cancel__disj,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N) )
    <=> ( ( K_1 = zero_zero_nat )
        | ( M = N ) ) ) ).

tff(fact_2491_mult__cancel1,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N) )
    <=> ( ( M = N )
        | ( K_1 = zero_zero_nat ) ) ) ).

tff(fact_2492_mult__cancel2,axiom,
    ! [M: nat,K_1: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1) )
    <=> ( ( M = N )
        | ( K_1 = zero_zero_nat ) ) ) ).

tff(fact_2493_diffs0__imp__equal,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) = zero_zero_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1) = zero_zero_nat )
       => ( M_1 = N_1 ) ) ) ).

tff(fact_2494_diff__self__eq__0,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),M_1) = zero_zero_nat ).

tff(fact_2495_minus__nat_Odiff__0,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),zero_zero_nat) = M_1 ).

tff(fact_2496_diff__0__eq__0,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N_1) = zero_zero_nat ).

tff(fact_2497_nat__less__cases,axiom,
    ! [P_1: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),M)) )
     => ( ( ( M = N )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),M)) )
       => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),M)) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N),M)) ) ) ) ).

tff(fact_2498_less__not__refl3,axiom,
    ! [S_2: nat,T_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_2),T_2))
     => ( S_2 != T_2 ) ) ).

tff(fact_2499_less__not__refl2,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
     => ( M_1 != N_1 ) ) ).

tff(fact_2500_less__irrefl__nat,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).

tff(fact_2501_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).

tff(fact_2502_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) ) ) ).

tff(fact_2503_less__not__refl,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).

tff(fact_2504_diff__less__mono2,axiom,
    ! [L_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M_1))) ) ) ).

tff(fact_2505_less__imp__diff__less,axiom,
    ! [N_1: nat,J_2: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),N_1)),K_2)) ) ).

tff(fact_2506_diff__add__inverse2,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),N_1) = M_1 ).

tff(fact_2507_diff__add__inverse,axiom,
    ! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1)),N_1) = M_1 ).

tff(fact_2508_diff__diff__left,axiom,
    ! [I: nat,J_2: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2)) ).

tff(fact_2509_Nat_Odiff__cancel,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ).

tff(fact_2510_diff__cancel2,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ).

tff(fact_2511_nat__add__right__cancel,axiom,
    ! [M: nat,K_1: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1) )
    <=> ( M = N ) ) ).

tff(fact_2512_nat__add__left__cancel,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N) )
    <=> ( M = N ) ) ).

tff(fact_2513_nat__add__assoc,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_2)) ).

tff(fact_2514_nat__add__left__commute,axiom,
    ! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z_1)) ).

tff(fact_2515_nat__add__commute,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1) ).

tff(fact_2516_add__mult__distrib2,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).

tff(fact_2517_add__mult__distrib,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_2)) ).

tff(fact_2518_left__add__mult__distrib,axiom,
    ! [I: nat,U_1: nat,J_2: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),U_1)),K_2) ).

tff(fact_2519_Nat_Odiff__le__self,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),M_1)) ).

tff(fact_2520_diff__le__mono2,axiom,
    ! [L_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M_1))) ) ).

tff(fact_2521_diff__le__mono,axiom,
    ! [L_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),L_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),L_2))) ) ).

tff(fact_2522_diff__diff__cancel,axiom,
    ! [I: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I)) = I ) ) ).

tff(fact_2523_eq__diff__iff,axiom,
    ! [N: nat,K_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1) )
        <=> ( M = N ) ) ) ) ).

tff(fact_2524_Nat_Odiff__diff__eq,axiom,
    ! [N_1: nat,K_2: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ) ) ) ).

tff(fact_2525_le__diff__iff,axiom,
    ! [N: nat,K_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ) ).

tff(fact_2526_le__refl,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).

tff(fact_2527_nat__le__linear,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) ) ).

tff(fact_2528_eq__imp__le,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( M_1 = N_1 )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).

tff(fact_2529_le__trans,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2)) ) ) ).

tff(fact_2530_le__antisym,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
       => ( M_1 = N_1 ) ) ) ).

tff(fact_2531_mult__le__mono,axiom,
    ! [K_2: nat,L_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),L_2))) ) ) ).

tff(fact_2532_mult__le__mono2,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))) ) ).

tff(fact_2533_mult__le__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))) ) ).

tff(fact_2534_le__cube,axiom,
    ! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1)))) ).

tff(fact_2535_le__square,axiom,
    ! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1))) ).

tff(fact_2536_nat__mult__1,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N_1) = N_1 ).

tff(fact_2537_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

tff(fact_2538_nat__mult__1__right,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),one_one_nat) = N_1 ).

tff(fact_2539_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

tff(fact_2540_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

tff(fact_2541_mult__less__cancel1,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_2542_mult__less__cancel2,axiom,
    ! [M: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_2543_nat__mult__eq__cancel1,axiom,
    ! [M: nat,N: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N) )
      <=> ( M = N ) ) ) ).

tff(fact_2544_nat__mult__less__cancel1,axiom,
    ! [M: nat,N: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_2545_mult__less__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))) ) ) ).

tff(fact_2546_mult__less__mono2,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))) ) ) ).

tff(fact_2547_gr0I,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_2548_gr__implies__not0,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( N_1 != zero_zero_nat ) ) ).

tff(fact_2549_less__nat__zero__code,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).

tff(fact_2550_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_2551_not__less0,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).

tff(fact_2552_diff__less,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),M_1)) ) ) ).

tff(fact_2553_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2554_plus__nat_Oadd__0,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N_1) = N_1 ).

tff(fact_2555_Nat_Oadd__0__right,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),zero_zero_nat) = M_1 ).

tff(fact_2556_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

tff(fact_2557_add__eq__self__zero,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = M_1 )
     => ( N_1 = zero_zero_nat ) ) ).

tff(fact_2558_diff__add__0,axiom,
    ! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1)) = zero_zero_nat ).

tff(fact_2559_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_2560_diff__is__0__eq_H,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) = zero_zero_nat ) ) ).

tff(fact_2561_less__eq__nat_Osimps_I1_J,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).

tff(fact_2562_le__0__eq,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

tff(fact_2563_not__add__less1,axiom,
    ! [I: nat,J_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),I)) ).

tff(fact_2564_not__add__less2,axiom,
    ! [J_2: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),I)) ).

tff(fact_2565_nat__add__left__cancel__less,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2566_trans__less__add1,axiom,
    ! [M_1: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M_1))) ) ).

tff(fact_2567_trans__less__add2,axiom,
    ! [M_1: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J_2))) ) ).

tff(fact_2568_add__less__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))) ) ).

tff(fact_2569_add__less__mono,axiom,
    ! [K_2: nat,L_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2))) ) ) ).

tff(fact_2570_less__add__eq__less,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat,L_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),L_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).

tff(fact_2571_add__lessD1,axiom,
    ! [I: nat,J_2: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) ) ).

tff(fact_2572_add__diff__inverse,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = M_1 ) ) ).

tff(fact_2573_less__diff__conv,axiom,
    ! [I_1: nat,J: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) ) ).

tff(fact_2574_mult__eq__self__implies__10,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( M_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) )
     => ( ( N_1 = one_one_nat )
        | ( M_1 = zero_zero_nat ) ) ) ).

tff(fact_2575_less__or__eq__imp__le,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
        | ( M_1 = N_1 ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).

tff(fact_2576_le__neq__implies__less,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => ( ( M_1 != N_1 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).

tff(fact_2577_less__imp__le__nat,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).

tff(fact_2578_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

tff(fact_2579_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

tff(fact_2580_less__diff__iff,axiom,
    ! [N: nat,K_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ) ).

tff(fact_2581_diff__less__mono,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_1),C_1))) ) ) ).

tff(fact_2582_diff__diff__right,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),J_2) ) ) ).

tff(fact_2583_le__diff__conv,axiom,
    ! [J: nat,K_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))) ) ).

tff(fact_2584_le__add__diff,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1)),K_2))) ) ).

tff(fact_2585_le__add__diff__inverse,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = M_1 ) ) ).

tff(fact_2586_add__diff__assoc,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) ) ) ).

tff(fact_2587_le__diff__conv2,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) ) ) ).

tff(fact_2588_le__add__diff__inverse2,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1) = M_1 ) ) ).

tff(fact_2589_le__imp__diff__is__add,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1) = K_1 )
      <=> ( J = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) ) ) ) ).

tff(fact_2590_diff__add__assoc,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) ) ) ).

tff(fact_2591_add__diff__assoc2,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) ) ) ).

tff(fact_2592_diff__add__assoc2,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) ) ) ).

tff(fact_2593_nat__eq__add__iff2,axiom,
    ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N) ) ) ) ).

tff(fact_2594_nat__diff__add__eq2,axiom,
    ! [U_1: nat,M_1: nat,N_1: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I)),U_1)),N_1)) ) ) ).

tff(fact_2595_nat__le__add__iff2,axiom,
    ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N))) ) ) ).

tff(fact_2596_nat__eq__add__iff1,axiom,
    ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),M) = N ) ) ) ).

tff(fact_2597_nat__diff__add__eq1,axiom,
    ! [U_1: nat,M_1: nat,N_1: nat,J_2: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),U_1)),M_1)),N_1) ) ) ).

tff(fact_2598_nat__le__add__iff1,axiom,
    ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),M)),N)) ) ) ).

tff(fact_2599_add__leE,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)),N_1))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) ) ) ).

tff(fact_2600_add__leD1,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).

tff(fact_2601_add__leD2,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) ) ).

tff(fact_2602_add__le__mono,axiom,
    ! [K_2: nat,L_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2))) ) ) ).

tff(fact_2603_add__le__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))) ) ).

tff(fact_2604_trans__le__add2,axiom,
    ! [M_1: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J_2))) ) ).

tff(fact_2605_trans__le__add1,axiom,
    ! [M_1: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M_1))) ) ).

tff(fact_2606_nat__add__left__cancel__le,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_2607_le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
    <=> ? [K: nat] : N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K) ) ).

tff(fact_2608_le__Suc__ex__iff,axiom,
    ! [K_1: nat,L: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
    <=> ? [N_2: nat] : L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_2) ) ).

tff(fact_2609_le__add1,axiom,
    ! [N_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1))) ).

tff(fact_2610_le__add2,axiom,
    ! [N_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1))) ).

tff(fact_2611_eq__diff__eq_H,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),Z_2) )
    <=> ( Y_2 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Z_2) ) ) ).

tff(fact_2612_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

tff(fact_2613_nat__diff__split,axiom,
    ! [P_1: fun_nat_bool,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
        & ! [D_2: nat] :
            ( ( Aa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),D_2) )
           => hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).

tff(fact_2614_nat__diff__split__asm,axiom,
    ! [P_1: fun_nat_bool,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)))
    <=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
            & ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
          | ? [D_2: nat] :
              ( ( Aa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),D_2) )
              & ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).

tff(fact_2615_nat__mult__le__cancel1,axiom,
    ! [M: nat,N: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).

tff(fact_2616_mult__le__cancel2,axiom,
    ! [M: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).

tff(fact_2617_ex__least__nat__less,axiom,
    ! [N: nat,P_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(P_1,N))
       => ? [K: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N))
            & ! [I_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K))
               => ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
            & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ).

tff(fact_2618_nat__mod__eq__lemma,axiom,
    ! [X: nat,N_1: nat,Y: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(X),N_1) = hAPP_nat_nat(div_mod_nat(Y),N_1) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y),X))
       => ? [Q_3: nat] : X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_3)) ) ) ).

tff(fact_2619_mod__eqD,axiom,
    ! [M_1: nat,D_3: nat,R_3: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M_1),D_3) = R_3 )
     => ? [Q_3: nat] : M_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_3),D_3)) ) ).

tff(fact_2620_nat__mod__eq__iff,axiom,
    ! [X_1: nat,N: nat,Y_2: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(X_1),N) = hAPP_nat_nat(div_mod_nat(Y_2),N) )
    <=> ? [Q1: nat,Q2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q2)) ) ).

tff(fact_2621_mod__eq__0D,axiom,
    ! [M_6: nat,D_4: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M_6),D_4) = zero_zero_nat )
     => ? [Q_3: nat] : M_6 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_3) ) ).

tff(fact_2622_mono__nat__linear__lb,axiom,
    ! [M: nat,K_1: nat,F: fun_nat_nat] :
      ( ! [M_2: nat,N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),hAPP_nat_nat(F,N_2))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,M)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_1)))) ) ).

tff(fact_2623_ex__least__nat__le,axiom,
    ! [N: nat,P_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(P_1,N))
       => ? [K: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
            & ! [I_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K))
               => ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
            & hBOOL(hAPP_nat_bool(P_1,K)) ) ) ) ).

tff(fact_2624_less__imp__add__positive,axiom,
    ! [I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ? [K: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
          & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K) = J_2 ) ) ) ).

tff(fact_2625_mult__left__cancel,axiom,
    ! [N_1: nat,M_1: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1) )
       => ( N_1 = M_1 ) ) ) ).

tff(fact_2626_divmod__nat__rel__mult1__eq,axiom,
    ! [A_2: nat,B_1: nat,C_1: nat,Q_1: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_1,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_3)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
       => hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),Q_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),R_3)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),R_3)),C_1)))) ) ) ).

tff(fact_2627_divmod__nat__rel__mult2__eq,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat,Q_1: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_2,B_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_3)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
         => hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_2,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Q_1),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),hAPP_nat_nat(div_mod_nat(Q_1),C_1))),R_3)))) ) ) ) ).

tff(fact_2628_lemma__interval,axiom,
    ! [B_1: real,A_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_1))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [Y_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y_1))),D_2))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),B_1)) ) ) ) ) ) ).

tff(fact_2629_le__Suc__ex,axiom,
    ! [K_2: nat,L_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
     => ? [N_2: nat] : L_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2) ) ).

tff(fact_2630_less__mono__imp__le__mono,axiom,
    ! [I_1: nat,J: nat,F: fun_nat_nat] :
      ( ! [I_2: nat,J_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J))) ) ) ).

tff(fact_2631_real__lbound__gt__zero,axiom,
    ! [D2: real,D1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2)) ) ) ) ).

tff(fact_2632_natceiling__add__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),one_one_nat) ) ) ).

tff(fact_2633_divmod__nat__rel__unique,axiom,
    ! [Qr_2: product_prod_nat_nat,M_1: nat,N_1: nat,Qr_1: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),Qr_1))
     => ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),Qr_2))
       => ( Qr_1 = Qr_2 ) ) ) ).

tff(fact_2634_natceiling__number__of__eq,axiom,
    ! [N_1: int] : natceiling(hAPP_int_real(number267125858f_real,N_1)) = hAPP_int_nat(number_number_of_nat,N_1) ).

tff(fact_2635_div__eq,axiom,
    ! [M_1: nat,N_1: nat,Q_1: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_3)))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = Q_1 ) ) ).

tff(fact_2636_mod__eq,axiom,
    ! [M_1: nat,N_1: nat,Q_1: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_3)))
     => ( hAPP_nat_nat(div_mod_nat(M_1),N_1) = R_3 ) ) ).

tff(fact_2637_natceiling__zero,axiom,
    natceiling(zero_zero_real) = zero_zero_nat ).

tff(fact_2638_zero__le__natceiling,axiom,
    ! [X: real] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X))) ).

tff(fact_2639_natceiling__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),natceiling(Y))) ) ).

tff(fact_2640_natceiling__one,axiom,
    natceiling(one_one_real) = one_one_nat ).

tff(fact_2641_divmod__nat__rel,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)),hAPP_nat_nat(div_mod_nat(M_1),N_1)))) ).

tff(fact_2642_natceiling__neg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( natceiling(X) = zero_zero_nat ) ) ).

tff(fact_2643_natceiling__le__eq__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),one_one_nat))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).

tff(fact_2644_divmod__nat__rel__add1__eq,axiom,
    ! [B_1: nat,Bq: nat,Br: nat,A_2: nat,C_1: nat,Aq: nat,Ar: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aq),Ar)))
     => ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_1,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Bq),Br)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
         => hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),B_1),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aq),Bq)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1)))) ) ) ) ).

tff(fact_2645_lemma__interval__lt,axiom,
    ! [B_1: real,A_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_1))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [Y_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y_1))),D_2))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_1)) ) ) ) ) ) ).

tff(fact_2646_divmod__nat__rel__ex,axiom,
    ! [M_1: nat,N_1: nat] :
      ~ ! [Q_3: nat,R_1: nat] : ~ hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_3),R_1))) ).

tff(fact_2647_natfloor__add__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),one_one_nat) ) ) ).

tff(fact_2648_nat__mult__eq__one,axiom,
    ! [N: nat,M: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) = one_one_nat )
    <=> ( ( N = one_one_nat )
        & ( M = one_one_nat ) ) ) ).

tff(fact_2649_natceiling__eq,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real)))
       => ( natceiling(X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat) ) ) ) ).

tff(fact_2650_natceiling__add__number__of,axiom,
    ! [X: real,N_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),hAPP_int_nat(number_number_of_nat,N_1)) ) ) ) ).

tff(fact_2651_divmod__int__def,axiom,
    ! [B_1: int,A_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1))
           => ( divmod_int(A_2,B_1) = posDivAlg(A_2,B_1) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1))
           => ( ( ( A_2 = zero_zero_int )
               => ( divmod_int(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ) )
              & ( ( A_2 != zero_zero_int )
               => ( divmod_int(A_2,B_1) = hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1))) ) ) ) ) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
           => ( divmod_int(A_2,B_1) = negDivAlg(A_2,B_1) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
           => ( divmod_int(A_2,B_1) = hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1))) ) ) ) ) ) ).

tff(fact_2652_divmod__int__mod__div,axiom,
    ! [P_3: int,Q_1: int] : divmod_int(P_3,Q_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_1)),hAPP_int_int(div_mod_int(P_3),Q_1)) ).

tff(fact_2653_real__of__nat__ge__zero,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N_1))) ).

tff(fact_2654_natfloor__real__of__nat,axiom,
    ! [N_1: nat] : natfloor(hAPP_nat_real(real_nat,N_1)) = N_1 ).

tff(fact_2655_Ints__real__of__nat,axiom,
    ! [N: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N)),ring_1_Ints_real)) ).

tff(fact_2656_natfloor__number__of__eq,axiom,
    ! [N_1: int] : natfloor(hAPP_int_real(number267125858f_real,N_1)) = hAPP_int_nat(number_number_of_nat,N_1) ).

tff(fact_2657_real__of__nat__inject,axiom,
    ! [N: nat,M: nat] :
      ( ( hAPP_nat_real(real_nat,N) = hAPP_nat_real(real_nat,M) )
    <=> ( N = M ) ) ).

tff(fact_2658_real__of__nat__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_real ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_real(number267125858f_real,V_2) ) ) ) ).

tff(fact_2659_real__natfloor__le,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X))),X)) ) ).

tff(fact_2660_le__natfloor,axiom,
    ! [X: nat,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X)),A_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),natfloor(A_2))) ) ).

tff(fact_2661_abs__real__of__nat__cancel,axiom,
    ! [X: nat] : hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X) ).

tff(fact_2662_not__neg__0,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).

tff(fact_2663_not__neg__1,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).

tff(fact_2664_natceiling__real__of__nat,axiom,
    ! [N_1: nat] : natceiling(hAPP_nat_real(real_nat,N_1)) = N_1 ).

tff(fact_2665_le__natfloor__eq,axiom,
    ! [Aa: nat,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),natfloor(X_1)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Aa)),X_1)) ) ) ).

tff(fact_2666_real__natfloor__add__one__gt,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,natfloor(X))),one_one_real))) ).

tff(fact_2667_natfloor__subtract,axiom,
    ! [A_2: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_2)),X))
     => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X)),A_2) ) ) ).

tff(fact_2668_real__natfloor__gt__diff__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X)))) ).

tff(fact_2669_real__of__nat__zero,axiom,
    hAPP_nat_real(real_nat,zero_zero_nat) = zero_zero_real ).

tff(fact_2670_real__of__nat__zero__iff,axiom,
    ! [N: nat] :
      ( ( hAPP_nat_real(real_nat,N) = zero_zero_real )
    <=> ( N = zero_zero_nat ) ) ).

tff(fact_2671_not__real__of__nat__less__zero,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real)) ).

tff(fact_2672_real__of__nat__less__iff,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) ) ).

tff(fact_2673_real__of__nat__le__iff,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).

tff(fact_2674_less__natfloor,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N_1)) ) ) ).

tff(fact_2675_real__of__nat__mult,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_1)) ).

tff(fact_2676_natfloor__add,axiom,
    ! [A_2: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),A_2) ) ) ).

tff(fact_2677_real__of__nat__add,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_1)) ).

tff(fact_2678_ge__natfloor__plus__one__imp__gt,axiom,
    ! [Z_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(Z_1)),one_one_nat)),N_1))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N_1))) ) ).

tff(fact_2679_natfloor__eq,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real)))
       => ( natfloor(X) = N_1 ) ) ) ).

tff(fact_2680_real__of__nat__1,axiom,
    hAPP_nat_real(real_nat,one_one_nat) = one_one_real ).

tff(fact_2681_le__natfloor__eq__number__of,axiom,
    ! [X_1: real,N: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,N)),natfloor(X_1)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N)),X_1)) ) ) ) ).

tff(fact_2682_real__natceiling__ge,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X)))) ).

tff(fact_2683_neg__def,axiom,
    ! [Z_6: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,Z_6))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_6),zero_zero_int)) ) ).

tff(fact_2684_not__neg__eq__ge__0,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).

tff(fact_2685_neg__imp__number__of__eq__0,axiom,
    ! [V_2: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
     => ( hAPP_int_nat(number_number_of_nat,V_2) = zero_zero_nat ) ) ).

tff(fact_2686_neg__number__of__Min,axiom,
    hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,min))) ).

tff(fact_2687_natfloor__zero,axiom,
    natfloor(zero_zero_real) = zero_zero_nat ).

tff(fact_2688_zero__le__natfloor,axiom,
    ! [X: real] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X))) ).

tff(fact_2689_natfloor__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))) ) ).

tff(fact_2690_natfloor__one,axiom,
    natfloor(one_one_real) = one_one_nat ).

tff(fact_2691_natfloor__add__number__of,axiom,
    ! [X: real,N_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),hAPP_int_nat(number_number_of_nat,N_1)) ) ) ) ).

tff(fact_2692_real__of__nat__le__zero__cancel__iff,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),zero_zero_real))
    <=> ( N = zero_zero_nat ) ) ).

tff(fact_2693_real__of__nat__diff,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_1)) ) ) ).

tff(fact_2694_natceiling__le,axiom,
    ! [X: real,A_2: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A_2)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A_2)) ) ).

tff(fact_2695_eq__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( ( hAPP_int_nat(number_number_of_nat,V) = hAPP_int_nat(number_number_of_nat,V_1) )
    <=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V_1)),zero_zero_int)) )
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
         => ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
             => ( hAPP_int_int(number_number_of_int,V) = zero_zero_int ) )
            & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
             => ( V = V_1 ) ) ) ) ) ) ).

tff(fact_2696_nat__number__of__add__left,axiom,
    ! [V_3: int,K_2: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3))),K_2) ) ) ) ) ) ).

tff(fact_2697_natfloor__neg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( natfloor(X) = zero_zero_nat ) ) ).

tff(fact_2698_real__of__nat__gt__zero__cancel__iff,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_2699_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)),hAPP_nat_real(real_nat,M))) ) ).

tff(fact_2700_negateSnd__eq,axiom,
    ! [Q_1: int,R_3: int] : hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),hAPP_int_int(uminus_uminus_int,R_3)) ).

tff(fact_2701_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M)),one_one_real))) ) ).

tff(fact_2702_natceiling__le__eq,axiom,
    ! [Aa: nat,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),Aa))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,Aa))) ) ) ).

tff(fact_2703_divmod__int__rel__neg,axiom,
    ! [A_2: int,B_1: int,Qr_1: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)),Qr_1))
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),hAPP_P1975530577nt_int(negateSnd,Qr_1))) ) ).

tff(fact_2704_natceiling__subtract,axiom,
    ! [A_2: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_2)),X))
     => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X)),A_2) ) ) ).

tff(fact_2705_divmod__int__correct,axiom,
    ! [A_2: int,B_1: int] :
      ( ( B_1 != zero_zero_int )
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_2,B_1),divmod_int(A_2,B_1))) ) ).

tff(fact_2706_le__natfloor__eq__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ).

tff(fact_2707_natceiling__add,axiom,
    ! [A_2: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),A_2) ) ) ).

tff(fact_2708_natceiling__le__eq__number__of,axiom,
    ! [X_1: real,N: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),hAPP_int_nat(number_number_of_nat,N)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(number267125858f_real,N))) ) ) ) ).

tff(fact_2709_le__mult__natfloor,axiom,
    ! [B_1: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,natfloor(A_2)),natfloor(B_1))),natfloor(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),B_1)))) ) ) ).

tff(fact_2710_reals__Archimedean6,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
     => ? [N_2: nat] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),one_one_nat))),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,N_2))) ) ) ).

tff(fact_2711_reals__Archimedean3,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y_1: real] :
        ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),X))) ) ).

tff(fact_2712_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_2))) ).

tff(fact_2713_div__pos__neg__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) ) ) ).

tff(fact_2714_mod__pos__neg__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int))
     => ( hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) ) ) ).

tff(fact_2715_natfloor__div__nat,axiom,
    ! [Y: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y))
       => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_nat_real(real_nat,Y))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X)),Y) ) ) ) ).

tff(fact_2716_div__neg__neg,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)))) ) ) ) ).

tff(fact_2717_mod__neg__neg,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)))) ) ) ) ).

tff(fact_2718_prod__eqI,axiom,
    ! [P_6: produc914805421l_real,Q_4: produc914805421l_real] :
      ( ( hAPP_P731461727l_real(produc1935615926l_real,P_6) = hAPP_P731461727l_real(produc1935615926l_real,Q_4) )
     => ( ( hAPP_P731461727l_real(produc556554744l_real,P_6) = hAPP_P731461727l_real(produc556554744l_real,Q_4) )
       => ( P_6 = Q_4 ) ) ) ).

tff(fact_2719_prod__eqI,axiom,
    ! [P_6: product_prod_nat_nat,Q_4: product_prod_nat_nat] :
      ( ( hAPP_P1654798560at_nat(product_fst_nat_nat,P_6) = hAPP_P1654798560at_nat(product_fst_nat_nat,Q_4) )
     => ( ( hAPP_P1654798560at_nat(product_snd_nat_nat,P_6) = hAPP_P1654798560at_nat(product_snd_nat_nat,Q_4) )
       => ( P_6 = Q_4 ) ) ) ).

tff(fact_2720_prod__eqI,axiom,
    ! [P_6: product_prod_int_int,Q_4: product_prod_int_int] :
      ( ( hAPP_P1175774780nt_int(product_fst_int_int,P_6) = hAPP_P1175774780nt_int(product_fst_int_int,Q_4) )
     => ( ( hAPP_P1175774780nt_int(product_snd_int_int,P_6) = hAPP_P1175774780nt_int(product_snd_int_int,Q_4) )
       => ( P_6 = Q_4 ) ) ) ).

tff(fact_2721_prod__eq__iff,axiom,
    ! [S_1: produc914805421l_real,T: produc914805421l_real] :
      ( ( S_1 = T )
    <=> ( ( hAPP_P731461727l_real(produc1935615926l_real,S_1) = hAPP_P731461727l_real(produc1935615926l_real,T) )
        & ( hAPP_P731461727l_real(produc556554744l_real,S_1) = hAPP_P731461727l_real(produc556554744l_real,T) ) ) ) ).

tff(fact_2722_prod__eq__iff,axiom,
    ! [S_1: product_prod_nat_nat,T: product_prod_nat_nat] :
      ( ( S_1 = T )
    <=> ( ( hAPP_P1654798560at_nat(product_fst_nat_nat,S_1) = hAPP_P1654798560at_nat(product_fst_nat_nat,T) )
        & ( hAPP_P1654798560at_nat(product_snd_nat_nat,S_1) = hAPP_P1654798560at_nat(product_snd_nat_nat,T) ) ) ) ).

tff(fact_2723_prod__eq__iff,axiom,
    ! [S_1: product_prod_int_int,T: product_prod_int_int] :
      ( ( S_1 = T )
    <=> ( ( hAPP_P1175774780nt_int(product_fst_int_int,S_1) = hAPP_P1175774780nt_int(product_fst_int_int,T) )
        & ( hAPP_P1175774780nt_int(product_snd_int_int,S_1) = hAPP_P1175774780nt_int(product_snd_int_int,T) ) ) ) ).

tff(fact_2724_surjective__pairing,axiom,
    ! [T_4: produc914805421l_real] : T_4 = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,T_4)),hAPP_P731461727l_real(produc556554744l_real,T_4)) ).

tff(fact_2725_surjective__pairing,axiom,
    ! [T_4: product_prod_nat_nat] : T_4 = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,T_4)),hAPP_P1654798560at_nat(product_snd_nat_nat,T_4)) ).

tff(fact_2726_surjective__pairing,axiom,
    ! [T_4: product_prod_int_int] : T_4 = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_fst_int_int,T_4)),hAPP_P1175774780nt_int(product_snd_int_int,T_4)) ).

tff(fact_2727_surjective__pairing,axiom,
    ! [T: produc1450482951t_bool] : T = hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,hAPP_P1477980390t_bool(produc926076176t_bool,T)),hAPP_P1477980390t_bool(produc840788306t_bool,T)) ).

tff(fact_2728_surjective__pairing,axiom,
    ! [T_4: produc1137372701nt_int] : T_4 = hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,hAPP_P141737927nt_int(produc1096438694nt_int,T_4)),hAPP_P141737927nt_int(produc1011150824nt_int,T_4)) ).

tff(fact_2729_surjective__pairing,axiom,
    ! [T_4: produc167071911de_int] : T_4 = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_P1913525324de_int(produc1968959274de_int,T_4)),hAPP_P1913525324de_int(produc221809260de_int,T_4)) ).

tff(fact_2730_surjective__pairing,axiom,
    ! [T_4: produc1359518119umeral] : T_4 = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_P619277728umeral(produc310214954umeral,T_4)),hAPP_P619277728umeral(produc1857364076umeral,T_4)) ).

tff(fact_2731_surjective__pairing,axiom,
    ! [T_4: produc1322466333at_nat] : T_4 = hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_P1502442695at_nat(produc707142054at_nat,T_4)),hAPP_P1502442695at_nat(produc621854184at_nat,T_4)) ).

tff(fact_2732_surjective__pairing,axiom,
    ! [T_4: produc393999548nt_int] : T_4 = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,hAPP_P1251258751nt_int(produc2110120675nt_int,T_4)),hAPP_P779853134nt_int(produc8341793nt_int,T_4)) ).

tff(fact_2733_pair__collapse,axiom,
    ! [P_5: produc914805421l_real] : hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,P_5)),hAPP_P731461727l_real(produc556554744l_real,P_5)) = P_5 ).

tff(fact_2734_pair__collapse,axiom,
    ! [P_5: product_prod_nat_nat] : hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,P_5)),hAPP_P1654798560at_nat(product_snd_nat_nat,P_5)) = P_5 ).

tff(fact_2735_pair__collapse,axiom,
    ! [P_5: product_prod_int_int] : hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_fst_int_int,P_5)),hAPP_P1175774780nt_int(product_snd_int_int,P_5)) = P_5 ).

tff(fact_2736_pair__collapse,axiom,
    ! [Pa: produc1450482951t_bool] : hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,hAPP_P1477980390t_bool(produc926076176t_bool,Pa)),hAPP_P1477980390t_bool(produc840788306t_bool,Pa)) = Pa ).

tff(fact_2737_pair__collapse,axiom,
    ! [P_5: produc1137372701nt_int] : hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,hAPP_P141737927nt_int(produc1096438694nt_int,P_5)),hAPP_P141737927nt_int(produc1011150824nt_int,P_5)) = P_5 ).

tff(fact_2738_pair__collapse,axiom,
    ! [P_5: produc167071911de_int] : hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_P1913525324de_int(produc1968959274de_int,P_5)),hAPP_P1913525324de_int(produc221809260de_int,P_5)) = P_5 ).

tff(fact_2739_pair__collapse,axiom,
    ! [P_5: produc1359518119umeral] : hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_P619277728umeral(produc310214954umeral,P_5)),hAPP_P619277728umeral(produc1857364076umeral,P_5)) = P_5 ).

tff(fact_2740_pair__collapse,axiom,
    ! [P_5: produc1322466333at_nat] : hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_P1502442695at_nat(produc707142054at_nat,P_5)),hAPP_P1502442695at_nat(produc621854184at_nat,P_5)) = P_5 ).

tff(fact_2741_pair__collapse,axiom,
    ! [P_5: produc393999548nt_int] : hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,hAPP_P1251258751nt_int(produc2110120675nt_int,P_5)),hAPP_P779853134nt_int(produc8341793nt_int,P_5)) = P_5 ).

tff(fact_2742_divide__zero__left,axiom,
    ! [A_10: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,zero_zero_rat),A_10) = zero_zero_rat ).

tff(fact_2743_divide__zero__left,axiom,
    ! [A_10: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,zero_zero_complex),A_10) = zero_zero_complex ).

tff(fact_2744_divide__zero__left,axiom,
    ! [A_10: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,zero_zero_real),A_10) = zero_zero_real ).

tff(fact_2745_divide__zero,axiom,
    ! [A_9: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,A_9),zero_zero_rat) = zero_zero_rat ).

tff(fact_2746_divide__zero,axiom,
    ! [A_9: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,A_9),zero_zero_complex) = zero_zero_complex ).

tff(fact_2747_divide__zero,axiom,
    ! [A_9: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_9),zero_zero_real) = zero_zero_real ).

tff(fact_2748_divide_Ozero,axiom,
    ! [Y_8: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,zero_zero_complex),Y_8) = zero_zero_complex ).

tff(fact_2749_divide_Ozero,axiom,
    ! [Y_8: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,zero_zero_real),Y_8) = zero_zero_real ).

tff(fact_2750_times__divide__times__eq,axiom,
    ! [X_7: rat,Y_7: rat,Z_5: rat,W_2: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X_7),Y_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Z_5),W_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_7),Z_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_7),W_2)) ).

tff(fact_2751_times__divide__times__eq,axiom,
    ! [X_7: complex,Y_7: complex,Z_5: complex,W_2: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_7),Y_7)),hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Z_5),W_2)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_7),Z_5)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_7),W_2)) ).

tff(fact_2752_times__divide__times__eq,axiom,
    ! [X_7: real,Y_7: real,Z_5: real,W_2: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_7),Y_7)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Z_5),W_2)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_7),Z_5)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_7),W_2)) ).

tff(fact_2753_times__divide__eq__right,axiom,
    ! [A_8: rat,B_8: rat,C_6: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_8),hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,B_8),C_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_8),B_8)),C_6) ).

tff(fact_2754_times__divide__eq__right,axiom,
    ! [A_8: complex,B_8: complex,C_6: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_8),hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,B_8),C_6)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_8),B_8)),C_6) ).

tff(fact_2755_times__divide__eq__right,axiom,
    ! [A_8: real,B_8: real,C_6: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_8),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,B_8),C_6)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_8),B_8)),C_6) ).

tff(fact_2756_add__divide__distrib,axiom,
    ! [A_7: rat,B_7: rat,C_5: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_7),B_7)),C_5) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,A_7),C_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,B_7),C_5)) ).

tff(fact_2757_add__divide__distrib,axiom,
    ! [A_7: complex,B_7: complex,C_5: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_7),B_7)),C_5) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,A_7),C_5)),hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,B_7),C_5)) ).

tff(fact_2758_add__divide__distrib,axiom,
    ! [A_7: real,B_7: real,C_5: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_7),B_7)),C_5) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_7),C_5)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,B_7),C_5)) ).

tff(fact_2759_divide_Oadd,axiom,
    ! [X_6: complex,Y_6: complex,Ya_2: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_6),Y_6)),Ya_2) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_6),Ya_2)),hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Y_6),Ya_2)) ).

tff(fact_2760_divide_Oadd,axiom,
    ! [X_6: real,Y_6: real,Ya_2: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_6),Y_6)),Ya_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_6),Ya_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_6),Ya_2)) ).

tff(fact_2761_divide__1,axiom,
    ! [A_6: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,A_6),one_one_rat) = A_6 ).

tff(fact_2762_divide__1,axiom,
    ! [A_6: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,A_6),one_one_complex) = A_6 ).

tff(fact_2763_divide__1,axiom,
    ! [A_6: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_6),one_one_real) = A_6 ).

tff(fact_2764_diff__divide__distrib,axiom,
    ! [A_5: rat,B_6: rat,C_4: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_5),B_6)),C_4) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,A_5),C_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,B_6),C_4)) ).

tff(fact_2765_diff__divide__distrib,axiom,
    ! [A_5: complex,B_6: complex,C_4: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_5),B_6)),C_4) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,A_5),C_4)),hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,B_6),C_4)) ).

tff(fact_2766_diff__divide__distrib,axiom,
    ! [A_5: real,B_6: real,C_4: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_5),B_6)),C_4) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_5),C_4)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,B_6),C_4)) ).

tff(fact_2767_divide_Odiff,axiom,
    ! [X_5: complex,Y_5: complex,Ya_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_5),Y_5)),Ya_1) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X_5),Ya_1)),hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Y_5),Ya_1)) ).

tff(fact_2768_divide_Odiff,axiom,
    ! [X_5: real,Y_5: real,Ya_1: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_5),Y_5)),Ya_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_5),Ya_1)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_5),Ya_1)) ).

tff(fact_2769_real__divide__square__eq,axiom,
    ! [R_3: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),R_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_2),R_3) ).

tff(fact_2770_real__0__le__divide__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
          | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) )
        & ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
          | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),zero_zero_real)) ) ) ) ).

tff(fact_2771_lemma__MVT,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Aa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa))),Aa)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Ba)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa))),Ba)) ).

tff(fact_2772_mod__int__def,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(div_mod_int(A_2),B_1) = hAPP_P1175774780nt_int(product_snd_int_int,divmod_int(A_2,B_1)) ).

tff(fact_2773_div__int__def,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = hAPP_P1175774780nt_int(product_fst_int_int,divmod_int(A_2,B_1)) ).

tff(fact_2774_real__of__nat__div4,axiom,
    ! [N_1: nat,X: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X)))) ).

tff(fact_2775_mod__neg__pos,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_2,B_1)) ) ) ) ).

tff(fact_2776_div__neg__pos,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = hAPP_P1175774780nt_int(product_fst_int_int,negDivAlg(A_2,B_1)) ) ) ) ).

tff(fact_2777_mod__pos__pos,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(A_2,B_1)) ) ) ) ).

tff(fact_2778_div__pos__pos,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(A_2,B_1)) ) ) ) ).

tff(fact_2779_real__of__nat__div2,axiom,
    ! [N_1: nat,X: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X))))) ).

tff(fact_2780_real__of__nat__div3,axiom,
    ! [N_1: nat,X: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),X)))),one_one_real)) ).

tff(fact_2781_mod__pos__pos__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) ) ) ).

tff(fact_2782_div__pos__pos__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) ) ) ).

tff(fact_2783_mod__pos__neg,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
       => ( hAPP_int_int(div_mod_int(A_2),B_1) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)))) ) ) ) ).

tff(fact_2784_real__of__nat__div__aux,axiom,
    ! [X: nat,D_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3))
     => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,X)),hAPP_nat_real(real_nat,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X),D_3))),hAPP_nat_real(real_nat,D_3))) ) ) ).

tff(fact_2785_div__pos__neg,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)))) ) ) ) ).

tff(fact_2786_arctan__add,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y)) = hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))) ) ) ) ).

tff(fact_2787_pair__lessI2,axiom,
    ! [S_1: nat,T: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T))
       => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),S_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Ba),T))),pair_less)) ) ) ).

tff(fact_2788_mod__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ) ) ).

tff(fact_2789_div__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ).

tff(fact_2790_divmod__nat__div__mod,axiom,
    ! [M_1: nat,N_1: nat] : divmod_nat(M_1,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1)),hAPP_nat_nat(div_mod_nat(M_1),N_1)) ).

tff(fact_2791_mod__nat__def,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(div_mod_nat(M_1),N_1) = hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M_1,N_1)) ).

tff(fact_2792_div__nat__def,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = hAPP_P1654798560at_nat(product_fst_nat_nat,divmod_nat(M_1,N_1)) ).

tff(fact_2793_arctan__minus,axiom,
    ! [X: real] : hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X)) ).

tff(fact_2794_nat__if__cong,axiom,
    ! [X_1: int,Y_2: int,P_1: bool] :
      ( ( hBOOL(P_1)
       => ( hAPP_int_nat(nat_1,X_1) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_1),Y_2)) ) )
      & ( ~ hBOOL(P_1)
       => ( hAPP_int_nat(nat_1,Y_2) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_1),Y_2)) ) ) ) ).

tff(fact_2795_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))) ) ).

tff(fact_2796_arctan__zero__zero,axiom,
    hAPP_real_real(arctan,zero_zero_real) = zero_zero_real ).

tff(fact_2797_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))) ) ).

tff(fact_2798_natceiling__def,axiom,
    ! [X: real] : natceiling(X) = hAPP_int_nat(nat_1,archim856651990g_real(X)) ).

tff(fact_2799_nat__0,axiom,
    hAPP_int_nat(nat_1,zero_zero_int) = zero_zero_nat ).

tff(fact_2800_transfer__nat__int__numerals_I1_J,axiom,
    zero_zero_nat = hAPP_int_nat(nat_1,zero_zero_int) ).

tff(fact_2801_eq__nat__nat__iff,axiom,
    ! [Z_3: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_3))
       => ( ( hAPP_int_nat(nat_1,Z_2) = hAPP_int_nat(nat_1,Z_3) )
        <=> ( Z_2 = Z_3 ) ) ) ) ).

tff(fact_2802_ex__nat,axiom,
    ! [P_1: fun_nat_bool] :
      ( ? [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
    <=> ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat_1,X_2))) ) ) ).

tff(fact_2803_all__nat,axiom,
    ! [P_1: fun_nat_bool] :
      ( ! [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
    <=> ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat_1,X_2))) ) ) ).

tff(fact_2804_transfer__nat__int__relations_I1_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( ( hAPP_int_nat(nat_1,X_1) = hAPP_int_nat(nat_1,Y_2) )
        <=> ( X_1 = Y_2 ) ) ) ) ).

tff(fact_2805_nat__number__of__def,axiom,
    ! [V_2: int] : hAPP_int_nat(number_number_of_nat,V_2) = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,V_2)) ).

tff(fact_2806_nat__number__of,axiom,
    ! [W: int] : hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,W)) = hAPP_int_nat(number_number_of_nat,W) ).

tff(fact_2807_transfer__nat__int__numerals_I2_J,axiom,
    one_one_nat = hAPP_int_nat(nat_1,one_one_int) ).

tff(fact_2808_neg__nat,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,Z_1))
     => ( hAPP_int_nat(nat_1,Z_1) = zero_zero_nat ) ) ).

tff(fact_2809_natfloor__def,axiom,
    ! [X: real] : natfloor(X) = hAPP_int_nat(nat_1,archim1246769320r_real(X)) ).

tff(fact_2810_divmod__nat__rel__divmod__nat,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),divmod_nat(M_1,N_1))) ).

tff(fact_2811_divmod__nat__eq,axiom,
    ! [M_1: nat,N_1: nat,Qr_1: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M_1,N_1),Qr_1))
     => ( divmod_nat(M_1,N_1) = Qr_1 ) ) ).

tff(fact_2812_nat__le__0,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int))
     => ( hAPP_int_nat(nat_1,Z_1) = zero_zero_nat ) ) ).

tff(fact_2813_nat__0__iff,axiom,
    ! [I_1: int] :
      ( ( hAPP_int_nat(nat_1,I_1) = zero_zero_nat )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) ) ).

tff(fact_2814_nat__mono__iff,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).

tff(fact_2815_zless__nat__conj,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).

tff(fact_2816_transfer__nat__int__relations_I3_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat_1,X_1)),hAPP_int_nat(nat_1,Y_2)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ) ) ).

tff(fact_2817_nat__abs__mult__distrib,axiom,
    ! [W: int,Z_1: int] : hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,W))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Z_1))) ).

tff(fact_2818_divmod__nat__rel__def,axiom,
    ! [M: nat,N: nat,Qr: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr))
    <=> ( ( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)),N)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)) )
        & ( ( N = zero_zero_nat )
         => ( hAPP_P1654798560at_nat(product_fst_nat_nat,Qr) = zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
             => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),N)) ) )
            & ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
             => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),zero_zero_nat)) ) ) ) ) ) ) ).

tff(fact_2819_zero__less__nat__eq,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat_1,Z_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).

tff(fact_2820_nat__less__eq__zless,axiom,
    ! [Z_2: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).

tff(fact_2821_transfer__nat__int__relations_I2_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,X_1)),hAPP_int_nat(nat_1,Y_2)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ) ) ).

tff(fact_2822_nat__le__eq__zle,axiom,
    ! [Z_2: int,W_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),W_1))
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ) ).

tff(fact_2823_nat__mult__distrib,axiom,
    ! [Z_4: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,Z_1)),hAPP_int_nat(nat_1,Z_4)) ) ) ).

tff(fact_2824_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) ) ) ) ).

tff(fact_2825_nat__add__distrib,axiom,
    ! [Z_4: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
       => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,Z_1)),hAPP_int_nat(nat_1,Z_4)) ) ) ) ).

tff(fact_2826_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) ) ) ) ).

tff(fact_2827_divmod__nat__zero,axiom,
    ! [M_1: nat] : divmod_nat(M_1,zero_zero_nat) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M_1) ).

tff(fact_2828_nat__diff__distrib,axiom,
    ! [Z_1: int,Z_4: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1))
       => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,Z_1)),hAPP_int_nat(nat_1,Z_4)) ) ) ) ).

tff(fact_2829_Divides_Otransfer__nat__int__functions_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) ) ) ) ).

tff(fact_2830_nat__div__distrib,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) ) ) ).

tff(fact_2831_Divides_Otransfer__nat__int__functions_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(X),Y)) ) ) ) ).

tff(fact_2832_nat__mod__distrib,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(X),Y)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) ) ) ) ).

tff(fact_2833_pair__lessI1,axiom,
    ! [S_1: nat,T: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
     => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),S_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Ba),T))),pair_less)) ) ).

tff(fact_2834_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(nat_tsub(X),Y)) ) ) ) ).

tff(fact_2835_nat__mult__distrib__neg,axiom,
    ! [Z_4: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int))
     => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(uminus_uminus_int,Z_1))),hAPP_int_nat(nat_1,hAPP_int_int(uminus_uminus_int,Z_4))) ) ) ).

tff(fact_2836_divmod__nat__base,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( divmod_nat(M_1,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M_1) ) ) ).

tff(fact_2837_nat__aux__def,axiom,
    ! [I: int,N_1: nat] : hAPP_nat_nat(nat_aux(I),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,I)),N_1) ).

tff(fact_2838_transfer__morphism__nat__int,axiom,
    hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(nat_1),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))) ).

tff(fact_2839_divmod__nat__step,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
       => ( divmod_nat(M_1,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1)) ) ) ) ).

tff(fact_2840_one__mod__nat__number__of,axiom,
    ! [V_3: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_nat_nat(suc,zero_zero_nat) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ).

tff(fact_2841_one__div__nat__number__of,axiom,
    ! [V_3: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,V_3))) ).

tff(fact_2842_arccos__lbound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y))) ) ) ).

tff(fact_2843_Suc__mono,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1))) ) ).

tff(fact_2844_lessI,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,N_1))) ).

tff(fact_2845_zero__less__Suc,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_1))) ).

tff(fact_2846_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

tff(fact_2847_nat_Oinject,axiom,
    ! [Nat_5: nat,Nat_4: nat] :
      ( ( hAPP_nat_nat(suc,Nat_5) = hAPP_nat_nat(suc,Nat_4) )
    <=> ( Nat_5 = Nat_4 ) ) ).

tff(fact_2848_Suc__n__not__n,axiom,
    ! [N_1: nat] : hAPP_nat_nat(suc,N_1) != N_1 ).

tff(fact_2849_n__not__Suc__n,axiom,
    ! [N_1: nat] : N_1 != hAPP_nat_nat(suc,N_1) ).

tff(fact_2850_Zero__not__Suc,axiom,
    ! [M_1: nat] : zero_zero_nat != hAPP_nat_nat(suc,M_1) ).

tff(fact_2851_nat_Osimps_I2_J,axiom,
    ! [Nat_3: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat_3) ).

tff(fact_2852_Suc__not__Zero,axiom,
    ! [M_1: nat] : hAPP_nat_nat(suc,M_1) != zero_zero_nat ).

tff(fact_2853_nat_Osimps_I3_J,axiom,
    ! [Nat_2: nat] : hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ).

tff(fact_2854_Zero__neq__Suc,axiom,
    ! [M_1: nat] : zero_zero_nat != hAPP_nat_nat(suc,M_1) ).

tff(fact_2855_Suc__neq__Zero,axiom,
    ! [M_1: nat] : hAPP_nat_nat(suc,M_1) != zero_zero_nat ).

tff(fact_2856_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M))) ) ).

tff(fact_2857_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

tff(fact_2858_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2859_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

tff(fact_2860_less__antisym,axiom,
    ! [N_1: nat,M_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M_1)))
       => ( M_1 = N_1 ) ) ) ).

tff(fact_2861_less__SucI,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1))) ) ).

tff(fact_2862_Suc__lessI,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ( ( hAPP_nat_nat(suc,M_1) != N_1 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N_1)) ) ) ).

tff(fact_2863_less__trans__Suc,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2)) ) ) ).

tff(fact_2864_less__SucE,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
       => ( M_1 = N_1 ) ) ) ).

tff(fact_2865_Suc__lessD,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).

tff(fact_2866_Suc__less__SucD,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).

tff(fact_2867_add__Suc__right,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) ).

tff(fact_2868_add__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) ).

tff(fact_2869_add__Suc__shift,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N_1)) ).

tff(fact_2870_Suc__leD,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).

tff(fact_2871_le__SucE,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_1)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
       => ( M_1 = hAPP_nat_nat(suc,N_1) ) ) ) ).

tff(fact_2872_le__SucI,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_1))) ) ).

tff(fact_2873_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).

tff(fact_2874_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
        | ( M = hAPP_nat_nat(suc,N) ) ) ) ).

tff(fact_2875_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M)) ) ).

tff(fact_2876_Suc__n__not__le__n,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1)) ).

tff(fact_2877_Suc__mult__cancel1,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N) )
    <=> ( M = N ) ) ).

tff(fact_2878_Suc__diff__diff,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1)),hAPP_nat_nat(suc,K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),K_2) ).

tff(fact_2879_diff__Suc__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ).

tff(fact_2880_mod__Suc__eq__Suc__mod,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M_1),N_1))),N_1) ).

tff(fact_2881_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J_1: nat] :
            ( ( M = hAPP_nat_nat(suc,J_1) )
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N)) ) ) ) ).

tff(fact_2882_less__Suc0,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

tff(fact_2883_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M_2: nat] : N = hAPP_nat_nat(suc,M_2) ) ).

tff(fact_2884_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_2885_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_2886_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
        & ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_2887_less__add__Suc1,axiom,
    ! [I: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M_1)))) ).

tff(fact_2888_less__add__Suc2,axiom,
    ! [I: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),I)))) ).

tff(fact_2889_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
    <=> ? [K: nat] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)) ) ).

tff(fact_2890_One__nat__def,axiom,
    one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_2891_Suc__le__lessD,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).

tff(fact_2892_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

tff(fact_2893_Suc__leI,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1)) ) ).

tff(fact_2894_le__imp__less__Suc,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1))) ) ).

tff(fact_2895_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2896_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_2897_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M)) ) ).

tff(fact_2898_Suc__mult__less__cancel1,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2899_diff__less__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),hAPP_nat_nat(suc,M_1))) ).

tff(fact_2900_mult__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) ).

tff(fact_2901_mult__Suc__right,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) ).

tff(fact_2902_Suc__mult__le__cancel1,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_2903_Suc__eq__plus1__left,axiom,
    ! [N_1: nat] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) ).

tff(fact_2904_Suc__eq__plus1,axiom,
    ! [N_1: nat] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat) ).

tff(fact_2905_Suc__diff__le,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) ) ) ).

tff(fact_2906_diff__Suc__1,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_1)),one_one_nat) = N_1 ).

tff(fact_2907_diff__Suc__eq__diff__pred,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1) ).

tff(fact_2908_div__1,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),hAPP_nat_nat(suc,zero_zero_nat)) = M_1 ).

tff(fact_2909_mod__Suc,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M_1),N_1)) = N_1 )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M_1)),N_1) = zero_zero_nat ) )
      & ( ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M_1),N_1)) != N_1 )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M_1),N_1)) ) ) ) ).

tff(fact_2910_mod__1,axiom,
    ! [M_1: nat] : hAPP_nat_nat(div_mod_nat(M_1),hAPP_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

tff(fact_2911_one__less__m,axiom,
    ! [M_1: nat,K_2: nat] :
      ( ( M_1 != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_2) )
     => ( ( M_1 != hAPP_nat_nat(suc,zero_zero_nat) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1)) ) ) ).

tff(fact_2912_one__less__k,axiom,
    ! [M_1: nat,K_2: nat] :
      ( ( M_1 != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_2) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_2)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2)) ) ) ).

tff(fact_2913_prod__mn__less__k,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
         => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = K_2 )
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),K_2)) ) ) ) ) ).

tff(fact_2914_mn__eq__m__one,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = M_1 )
       => ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_2915_n__less__m__mult__n,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1))) ) ) ).

tff(fact_2916_n__less__n__mult__m,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M_1))) ) ) ).

tff(fact_2917_one__less__mult,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1))) ) ) ).

tff(fact_2918_diff__Suc__less,axiom,
    ! [I: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,I))),N_1)) ) ).

tff(fact_2919_Suc__pred,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))) = N_1 ) ) ).

tff(fact_2920_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

tff(fact_2921_real__of__nat__Suc__gt__zero,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) ).

tff(fact_2922_nat__1,axiom,
    hAPP_int_nat(nat_1,one_one_int) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_2923_real__of__nat__one,axiom,
    hAPP_nat_real(real_nat,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_real ).

tff(fact_2924_diff__Suc__diff__eq1,axiom,
    ! [M_1: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)),hAPP_nat_nat(suc,J_2)) ) ) ).

tff(fact_2925_diff__Suc__diff__eq2,axiom,
    ! [M_1: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M_1)) ) ) ).

tff(fact_2926_real__of__nat__Suc,axiom,
    ! [N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real) ).

tff(fact_2927_mod__mult__self4,axiom,
    ! [K_2: nat,N_1: nat,M_1: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)),M_1))),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M_1)),N_1) ).

tff(fact_2928_Suc__diff__1,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) = N_1 ) ) ).

tff(fact_2929_Suc__pred_H,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) ) ) ).

tff(fact_2930_add__eq__if,axiom,
    ! [N_1: nat,M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = N_1 ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1)) ) ) ) ).

tff(fact_2931_div__geq,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1)) ) ) ) ).

tff(fact_2932_div__if,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = zero_zero_nat ) )
        & ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1)) ) ) ) ) ).

tff(fact_2933_Suc__times__mod__eq,axiom,
    ! [M_1: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),K_2))
     => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1))),K_2) = one_one_nat ) ) ).

tff(fact_2934_floor__eq3,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))))
       => ( hAPP_int_nat(nat_1,archim1246769320r_real(X)) = N_1 ) ) ) ).

tff(fact_2935_one__less__nat__eq,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(nat_1,Z_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Z_2)) ) ).

tff(fact_2936_expand__Suc,axiom,
    ! [V_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_2)))
     => ( hAPP_int_nat(number_number_of_nat,V_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)) ) ) ).

tff(fact_2937_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_nat_nat(suc,hAPP_int_nat(nat_1,Z_1)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)) ) ) ).

tff(fact_2938_split__div__lemma,axiom,
    ! [Qa: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Qa)),M))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(suc,Qa)))) )
      <=> ( Qa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) ) ) ) ).

tff(fact_2939_split__div_H,axiom,
    ! [P_1: fun_nat_bool,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
        | ? [Q_3: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_3)),M))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(suc,Q_3))))
            & hBOOL(hAPP_nat_bool(P_1,Q_3)) ) ) ) ).

tff(fact_2940_le__div__geq,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1)) ) ) ) ).

tff(fact_2941_floor__eq4,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))))
       => ( hAPP_int_nat(nat_1,archim1246769320r_real(X)) = N_1 ) ) ) ).

tff(fact_2942_reals__Archimedean4,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ? [N_2: nat] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),Y)),X))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2))),Y))) ) ) ) ).

tff(fact_2943_reals__Archimedean6a,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
     => ? [N_2: nat] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))) ) ) ).

tff(fact_2944_div__Suc,axiom,
    ! [A_2: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,A_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_2),C_1)),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),C_1))),C_1)) ).

tff(fact_2945_mod__induct__0,axiom,
    ! [I_1: nat,P_1: fun_nat_bool,Pa: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Pa))
         => ( hBOOL(hAPP_nat_bool(P_1,I_2))
           => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),Pa))) ) )
     => ( hBOOL(hAPP_nat_bool(P_1,I_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),Pa))
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ).

tff(fact_2946_mod__induct,axiom,
    ! [J: nat,I_1: nat,P_1: fun_nat_bool,Pa: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Pa))
         => ( hBOOL(hAPP_nat_bool(P_1,I_2))
           => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),Pa))) ) )
     => ( hBOOL(hAPP_nat_bool(P_1,I_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),Pa))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),Pa))
           => hBOOL(hAPP_nat_bool(P_1,J)) ) ) ) ) ).

tff(fact_2947_inc__induct,axiom,
    ! [P_1: fun_nat_bool,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(P_1,J))
       => ( ! [I_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J))
             => ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_2)))
               => hBOOL(hAPP_nat_bool(P_1,I_2)) ) )
         => hBOOL(hAPP_nat_bool(P_1,I_1)) ) ) ) ).

tff(fact_2948_nat__lt__two__imp__zero__or__one,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))
     => ( ( X = zero_zero_nat )
        | ( X = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_2949_less__imp__Suc__add,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
     => ? [K: nat] : N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K)) ) ).

tff(fact_2950_lemma__f__mono__add,axiom,
    ! [M: nat,No: nat,F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,M)),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),No)))) ) ).

tff(fact_2951_gr0__implies__Suc,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ? [M_2: nat] : N_1 = hAPP_nat_nat(suc,M_2) ) ).

tff(fact_2952_Suc__nat__number__of__add,axiom,
    ! [N_1: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))),N_1) ) ) ) ).

tff(fact_2953_zero__induct__lemma,axiom,
    ! [I_1: nat,P_1: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,K_1))
     => ( ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))
           => hBOOL(hAPP_nat_bool(P_1,N_2)) )
       => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_1))) ) ) ).

tff(fact_2954_Suc__le__D,axiom,
    ! [N_1: nat,M_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_5))
     => ? [M_2: nat] : M_5 = hAPP_nat_nat(suc,M_2) ) ).

tff(fact_2955_lessE,axiom,
    ! [I: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
     => ( ( K_2 != hAPP_nat_nat(suc,I) )
       => ~ ! [J_1: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
             => ( K_2 != hAPP_nat_nat(suc,J_1) ) ) ) ) ).

tff(fact_2956_Suc__lessE,axiom,
    ! [I: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2))
     => ~ ! [J_1: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
           => ( K_2 != hAPP_nat_nat(suc,J_1) ) ) ) ).

tff(fact_2957_not0__implies__Suc,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ? [M_2: nat] : N_1 = hAPP_nat_nat(suc,M_2) ) ).

tff(fact_2958_succ__def,axiom,
    ! [K_2: int] : hAPP_int_int(succ,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int) ).

tff(fact_2959_diff__bin__simps_I2_J,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),min) = hAPP_int_int(succ,K_2) ).

tff(fact_2960_Suc__nat__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ) ).

tff(fact_2961_exists__least__lemma,axiom,
    ! [P_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( ? [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
       => ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(P_1,N_2))
            & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) ) ) ) ).

tff(fact_2962_nat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat_1: nat] : Y != hAPP_nat_nat(suc,Nat_1) ) ).

tff(fact_2963_zero__induct,axiom,
    ! [P_1: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,K_1))
     => ( ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))
           => hBOOL(hAPP_nat_bool(P_1,N_2)) )
       => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ).

tff(fact_2964_nat__induct,axiom,
    ! [N: nat,P_1: fun_nat_bool] :
      ( hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(P_1,N_2))
           => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) )
       => hBOOL(hAPP_nat_bool(P_1,N)) ) ) ).

tff(fact_2965_int__power__div__base,axiom,
    ! [K_2: int,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),M_1)),K_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,zero_zero_nat))) ) ) ) ).

tff(fact_2966_pair__leqI2,axiom,
    ! [S_1: nat,T: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,S_1),T))
       => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),S_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Ba),T))),pair_leq)) ) ) ).

tff(fact_2967_zpower__zpower,axiom,
    ! [X: int,Y: nat,Z_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),Z_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1)) ).

tff(fact_2968_zpower__zmod,axiom,
    ! [X: int,M_1: int,Y: nat] : hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(div_mod_int(X),M_1)),Y)),M_1) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),M_1) ).

tff(fact_2969_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
    ! [N_1: nat,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N_1))) ) ).

tff(fact_2970_zpower__zadd__distrib,axiom,
    ! [X: int,Y: nat,Z_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)) ).

tff(fact_2971_zero__le__zpower__abs,axiom,
    ! [X: int,N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X)),N_1))) ).

tff(fact_2972_zero__less__zpower__abs__iff,axiom,
    ! [X_1: int,N: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),N)))
    <=> ( ( X_1 != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

tff(fact_2973_Euler_Oaux__1,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
     => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(nat_1,P_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,P_3)),one_one_nat))) ) ) ).

tff(fact_2974_pair__leqI1,axiom,
    ! [S_1: nat,T: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
     => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aa),S_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Ba),T))),pair_leq)) ) ).

tff(fact_2975_real__of__nat__power,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_1),N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M_1)),N_1) ).

tff(fact_2976_power__real__of__nat,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M_1)),N_1) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_1),N_1)) ).

tff(fact_2977_natfloor__power,axiom,
    ! [N_1: nat,X: real] :
      ( ( X = hAPP_nat_real(real_nat,natfloor(X)) )
     => ( natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X)),N_1) ) ) ).

tff(fact_2978_nat__power__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N) = zero_zero_nat )
    <=> ( ( N != zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

tff(fact_2979_exp__mono__eq,axiom,
    ! [X_1: nat,N: nat,Y_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N)) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_2980_nat__power__eq__Suc__0__iff,axiom,
    ! [X_1: nat,M: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),M) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_2981_power__Suc__0,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_2982_nat__zero__less__power__iff,axiom,
    ! [X_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1))
        | ( N = zero_zero_nat ) ) ) ).

tff(fact_2983_nat__power__less__imp__less,axiom,
    ! [M_1: nat,N_1: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).

tff(fact_2984_nat__one__le__power,axiom,
    ! [N_1: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),I))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1))) ) ).

tff(fact_2985_zero__less__power__nat__eq,axiom,
    ! [X_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N)))
    <=> ( ( N = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) ) ) ).

tff(fact_2986_exp__mono__lt,axiom,
    ! [X_1: nat,N: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N))))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ).

tff(fact_2987_exp__eq__1,axiom,
    ! [X_1: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N) = one_one_nat )
    <=> ( ( X_1 = one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

tff(fact_2988_exp__mono__le,axiom,
    ! [X_1: nat,N: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N))))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ).

tff(fact_2989_zero__less__power__nat__eq__number__of,axiom,
    ! [X_1: nat,W_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,W_1))))
    <=> ( ( hAPP_int_nat(number_number_of_nat,W_1) = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) ) ) ).

tff(fact_2990_power__eq__if,axiom,
    ! [P_3: nat,M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M_1) = one_one_nat ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat))) ) ) ) ).

tff(fact_2991_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
    ! [N_1: nat,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat_1,X)),N_1) = hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N_1)) ) ) ).

tff(fact_2992_nat__power__eq,axiom,
    ! [N_1: nat,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat_1,Z_1)),N_1) ) ) ).

tff(fact_2993_power__nat__number__of__number__of,axiom,
    ! [W: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W)) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_nat(number_number_of_nat,W))) ) ) ) ).

tff(fact_2994_power__nat__number__of,axiom,
    ! [N_1: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_1) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N_1) = hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),N_1)) ) ) ) ).

tff(fact_2995_LIMSEQ__inverse__realpow__zero__lemma,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X)),one_one_real)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)),N_1))) ) ).

tff(fact_2996_realpow__pos__nth,axiom,
    ! [A_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N_1) = A_2 ) ) ) ) ).

tff(fact_2997_realpow__pos__nth__unique,axiom,
    ! [A_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
       => ? [X_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
            & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N_1) = A_2 )
            & ! [Y_1: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
                  & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N_1) = A_2 ) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_2998_neg__one__power,axiom,
    ! [N_1: nat] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N_1) = one_one_int )
      | ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N_1) = hAPP_int_int(number_number_of_int,min) ) ) ).

tff(fact_2999_realpow__pos__nth2,axiom,
    ! [N_1: nat,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ? [R_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
          & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),hAPP_nat_nat(suc,N_1)) = A_2 ) ) ) ).

tff(fact_3000_EvenOdd_Oneg__one__even__power,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,X_1)) = one_one_int ) ) ) ).

tff(fact_3001_EvenOdd_Oneg__one__odd__power,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,X_1)) = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_3002_neg__one__power__parity,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
          <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) )
         => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,X_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,Y_2)) ) ) ) ) ).

tff(fact_3003_ceiling__eq2,axiom,
    ! [N_1: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
       => ( archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int) ) ) ) ).

tff(fact_3004_neg__odd__power,axiom,
    ! [Aa: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,Aa)),hAPP_int_nat(nat_1,X_1)) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Aa),hAPP_int_nat(nat_1,X_1))) ) ) ) ).

tff(fact_3005_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y)),pi)) ) ) ) ).

tff(fact_3006_even__odd__conj,axiom,
    ! [X_1: int] :
      ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
        & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_3007_odd__iff__not__even,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
    <=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_3008_even__odd__disj,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
      | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_3009_not__odd__impl__even,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_3010_odd__minus__odd,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven)) ) ) ).

tff(fact_3011_odd__minus__even,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zOdd)) ) ) ).

tff(fact_3012_even__minus__odd,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zOdd)) ) ) ).

tff(fact_3013_even__plus__odd__prop2,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),zOdd)) ) ) ).

tff(fact_3014_even__plus__odd__prop1,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),zEven)) ) ) ).

tff(fact_3015_IntNatAux_Oeven__plus__odd,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)),zOdd)) ) ) ).

tff(fact_3016_IntNatAux_Oodd__plus__odd,axiom,
    ! [Ba: int,Aa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)),zEven)) ) ) ).

tff(fact_3017_real__of__int__inject,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_real(real_int,X_1) = hAPP_int_real(real_int,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3018_pi__neq__zero,axiom,
    pi != zero_zero_real ).

tff(fact_3019_floor__real__of__int,axiom,
    ! [N_1: int] : archim1246769320r_real(hAPP_int_real(real_int,N_1)) = N_1 ).

tff(fact_3020_real__of__int__floor__cancel,axiom,
    ! [X_1: real] :
      ( ( hAPP_int_real(real_int,archim1246769320r_real(X_1)) = X_1 )
    <=> ? [N_2: int] : X_1 = hAPP_int_real(real_int,N_2) ) ).

tff(fact_3021_odd__minus__one__even,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),one_one_int)),zEven)) ) ).

tff(fact_3022_ceiling__real__of__int,axiom,
    ! [N_1: int] : archim856651990g_real(hAPP_int_real(real_int,N_1)) = N_1 ).

tff(fact_3023_real__of__int__ceiling__cancel,axiom,
    ! [X_1: real] :
      ( ( hAPP_int_real(real_int,archim856651990g_real(X_1)) = X_1 )
    <=> ? [N_2: int] : X_1 = hAPP_int_real(real_int,N_2) ) ).

tff(fact_3024_one__not__even,axiom,
    ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,one_one_int),zEven)) ).

tff(fact_3025_EvenOdd_Oeven__product,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
        | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) ) ) ).

tff(fact_3026_even__times__either,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zEven)) ) ).

tff(fact_3027_EvenOdd_Oeven__plus__even,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)),zEven)) ) ) ).

tff(fact_3028_even__minus__even,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven)) ) ) ).

tff(fact_3029_even__diff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) ) ) ).

tff(fact_3030_odd__mult__odd__prop,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zOdd))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) ) ).

tff(fact_3031_EvenOdd_Oodd__times__odd,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zOdd)) ) ) ).

tff(fact_3032_real__of__int__zero__cancel,axiom,
    ! [X_1: int] :
      ( ( hAPP_int_real(real_int,X_1) = zero_zero_real )
    <=> ( X_1 = zero_zero_int ) ) ).

tff(fact_3033_real__of__int__zero,axiom,
    hAPP_int_real(real_int,zero_zero_int) = zero_zero_real ).

tff(fact_3034_real__of__int__less__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_3035_real__of__int__le__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_3036_real__of__int__mult,axiom,
    ! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).

tff(fact_3037_real__of__one,axiom,
    hAPP_int_real(real_int,one_one_int) = one_one_real ).

tff(fact_3038_real__number__of,axiom,
    ! [V_2: int] : hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_2)) = hAPP_int_real(number267125858f_real,V_2) ).

tff(fact_3039_real__of__int__add,axiom,
    ! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).

tff(fact_3040_real__of__int__power,axiom,
    ! [X: int,N_1: nat] : hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N_1) ).

tff(fact_3041_power__real__of__int,axiom,
    ! [X: int,N_1: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N_1) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N_1)) ).

tff(fact_3042_real__of__int__diff,axiom,
    ! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).

tff(fact_3043_pi__ge__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

tff(fact_3044_pi__gt__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),pi)) ).

tff(fact_3045_pi__not__less__zero,axiom,
    ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),zero_zero_real)) ).

tff(fact_3046_real__of__int__minus,axiom,
    ! [X: int] : hAPP_int_real(real_int,hAPP_int_int(uminus_uminus_int,X)) = hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,X)) ).

tff(fact_3047_Ints__real__of__int,axiom,
    ! [X_1: int] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,X_1)),ring_1_Ints_real)) ).

tff(fact_3048_real__of__int__abs,axiom,
    ! [X: int] : hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X)) = hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X)) ).

tff(fact_3049_real__of__int__floor__le,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),R_3)) ).

tff(fact_3050_real__of__int__ceiling__ge,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_int_real(real_int,archim856651990g_real(R_3)))) ).

tff(fact_3051_real__of__int__div4,axiom,
    ! [N_1: int,X: int] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X)))) ).

tff(fact_3052_floor__less__eq,axiom,
    ! [X_1: real,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),Aa))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(real_int,Aa))) ) ).

tff(fact_3053_le__floor,axiom,
    ! [A_2: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_2)),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),archim1246769320r_real(X))) ) ).

tff(fact_3054_le__floor__eq,axiom,
    ! [Aa: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Aa)),X_1)) ) ).

tff(fact_3055_real__le__floor,axiom,
    ! [A_2: int,X: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),archim1246769320r_real(X)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_2)),X)) ) ).

tff(fact_3056_less__ceiling__eq,axiom,
    ! [Aa: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Aa)),X_1)) ) ).

tff(fact_3057_floor__add,axiom,
    ! [X: real,A_2: int] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_2))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X)),A_2) ).

tff(fact_3058_ceiling__le__eq,axiom,
    ! [X_1: real,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),Aa))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,Aa))) ) ).

tff(fact_3059_ceiling__le,axiom,
    ! [X: real,A_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_2)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_2)) ) ).

tff(fact_3060_ceiling__le__real,axiom,
    ! [X: real,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_2))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_2))) ) ).

tff(fact_3061_floor__power,axiom,
    ! [N_1: nat,X: real] :
      ( ( X = hAPP_int_real(real_int,archim1246769320r_real(X)) )
     => ( archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X)),N_1) ) ) ).

tff(fact_3062_floor__subtract,axiom,
    ! [X: real,A_2: int] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_2))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X)),A_2) ).

tff(fact_3063_ceiling__add,axiom,
    ! [X: real,A_2: int] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_2))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X)),A_2) ).

tff(fact_3064_floor__minus__real__of__int,axiom,
    ! [N_1: int] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N_1))) = hAPP_int_int(uminus_uminus_int,N_1) ).

tff(fact_3065_ceiling__subtract,axiom,
    ! [X: real,A_2: int] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_2))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X)),A_2) ).

tff(fact_3066_power__preserves__even,axiom,
    ! [X_1: int,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N)),zEven))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ) ).

tff(fact_3067_real__of__int__gt__zero__cancel__iff,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) ) ).

tff(fact_3068_real__of__int__lt__zero__cancel__iff,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),zero_zero_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),zero_zero_int)) ) ).

tff(fact_3069_real__of__int__le__zero__cancel__iff,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),zero_zero_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) ) ).

tff(fact_3070_real__of__int__ge__zero__cancel__iff,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) ) ).

tff(fact_3071_number__of__less__real__of__int__iff2,axiom,
    ! [M: int,N: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,M)),hAPP_int_real(number267125858f_real,N)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),hAPP_int_int(number_number_of_int,N))) ) ).

tff(fact_3072_number__of__less__real__of__int__iff,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,N)),hAPP_int_real(real_int,M)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,N)),M)) ) ).

tff(fact_3073_number__of__le__real__of__int__iff2,axiom,
    ! [M: int,N: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),hAPP_int_real(number267125858f_real,N)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),hAPP_int_int(number_number_of_int,N))) ) ).

tff(fact_3074_number__of__le__real__of__int__iff,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N)),hAPP_int_real(real_int,M)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,N)),M)) ) ).

tff(fact_3075_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),M))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)),hAPP_int_real(real_int,M))) ) ).

tff(fact_3076_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,M)),one_one_real))) ) ).

tff(fact_3077_lemma__floor2,axiom,
    ! [N_1: int,X: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),one_one_real)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),X)) ) ).

tff(fact_3078_real__nat__eq__real,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_nat_real(real_nat,hAPP_int_nat(nat_1,X)) = hAPP_int_real(real_int,X) ) ) ).

tff(fact_3079_power__preserves__odd,axiom,
    ! [X_1: int,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N)),zOdd))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) ) ) ).

tff(fact_3080_real__of__int__floor__add__one__ge,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real))) ).

tff(fact_3081_real__of__int__floor__add__one__gt,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real))) ).

tff(fact_3082_floor__eq,axiom,
    ! [N_1: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
       => ( archim1246769320r_real(X) = N_1 ) ) ) ).

tff(fact_3083_real__of__int__floor__ge__diff__one,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3)))) ).

tff(fact_3084_real__of__int__floor__gt__diff__one,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3)))) ).

tff(fact_3085_real__of__int__ceiling__le__add__one,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),one_one_real))) ).

tff(fact_3086_real__of__int__ceiling__diff__one__le,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),one_one_real)),R_3)) ).

tff(fact_3087_lemma__floor,axiom,
    ! [N_1: int,M_1: int,R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M_1)),R_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_1),N_1)) ) ) ).

tff(fact_3088_real__of__int__div2,axiom,
    ! [N_1: int,X: int] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X))))) ).

tff(fact_3089_real__of__int__div3,axiom,
    ! [N_1: int,X: int] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),X)))),one_one_real)) ).

tff(fact_3090_real__of__int__div__aux,axiom,
    ! [X: int,D_3: int] :
      ( ( D_3 != zero_zero_int )
     => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X),D_3))),hAPP_int_real(real_int,D_3))) ) ) ).

tff(fact_3091_floor__eq2,axiom,
    ! [N_1: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
       => ( archim1246769320r_real(X) = N_1 ) ) ) ).

tff(fact_3092_less__floor__eq,axiom,
    ! [Aa: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Aa)),one_one_real)),X_1)) ) ).

tff(fact_3093_floor__le__eq,axiom,
    ! [X_1: real,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),Aa))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Aa)),one_one_real))) ) ).

tff(fact_3094_ceiling__eq3,axiom,
    ! [N_1: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,N_1)),one_one_real)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N_1)))
       => ( archim856651990g_real(X) = N_1 ) ) ) ).

tff(fact_3095_ceiling__less__eq,axiom,
    ! [X_1: real,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),Aa))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,Aa)),one_one_real))) ) ).

tff(fact_3096_le__ceiling__eq,axiom,
    ! [Aa: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,Aa)),one_one_real)),X_1)) ) ).

tff(fact_3097_arccos__ubound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) ) ) ).

tff(fact_3098_ceiling__eq,axiom,
    ! [N_1: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)))
       => ( archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int) ) ) ) ).

tff(fact_3099_neg__even__power,axiom,
    ! [Aa: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,Aa)),hAPP_int_nat(nat_1,X_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Aa),hAPP_int_nat(nat_1,X_1)) ) ) ) ).

tff(fact_3100_arccos__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) ) ) ) ).

tff(fact_3101_reals__Archimedean__6c__int,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
     => ? [N_2: int] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ) ).

tff(fact_3102_reals__Archimedean__6b__int,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
     => ? [N_2: int] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ) ).

tff(fact_3103_real__lb__ub__int,axiom,
    ! [R_3: real] :
    ? [N_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ).

tff(fact_3104_arccos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi))
          & ( hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ) ) ).

tff(fact_3105_cos__npi2,axiom,
    ! [N_1: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N_1))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) ).

tff(fact_3106_cos__npi,axiom,
    ! [N_1: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) ).

tff(fact_3107_cos__minus,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(cos,X) ).

tff(fact_3108_cos__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),one_one_real)) ).

tff(fact_3109_cos__zero,axiom,
    hAPP_real_real(cos,zero_zero_real) = one_one_real ).

tff(fact_3110_cos__arctan__not__zero,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(arctan,X)) != zero_zero_real ).

tff(fact_3111_cos__monotone__0__pi_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3112_cos__ge__minus__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X))) ).

tff(fact_3113_abs__cos__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X))),one_one_real)) ).

tff(fact_3114_cos__pi,axiom,
    hAPP_real_real(cos,pi) = hAPP_int_real(number267125858f_real,min) ).

tff(fact_3115_cos__periodic__pi,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X)) ).

tff(fact_3116_cos__monotone__0__pi,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3117_cos__monotone__minus__pi__0_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))) ) ) ) ).

tff(fact_3118_arccos__cos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
       => ( hAPP_real_real(arccos,hAPP_real_real(cos,X)) = X ) ) ) ).

tff(fact_3119_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
     => ( hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ).

tff(fact_3120_cos__monotone__minus__pi__0,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))) ) ) ) ).

tff(fact_3121_cos__arccos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ) ).

tff(fact_3122_arccos__cos2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),X))
       => ( hAPP_real_real(arccos,hAPP_real_real(cos,X)) = hAPP_real_real(uminus_uminus_real,X) ) ) ) ).

tff(fact_3123_cos__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),pi))
            & ( hAPP_real_real(cos,X_2) = Y )
            & ! [Y_1: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),pi))
                  & ( hAPP_real_real(cos,Y_1) = Y ) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_3124_tan__add,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,Y) != zero_zero_real )
       => ( ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) != zero_zero_real )
         => ( hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)))) ) ) ) ) ).

tff(fact_3125_lemma__tan__add1,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,Y) != zero_zero_real )
       => ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3126_nat__1__add__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ) ).

tff(fact_3127_nat__number__of__add__1,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ) ).

tff(fact_3128_Pls__def,axiom,
    pls = zero_zero_int ).

tff(fact_3129_tan__zero,axiom,
    hAPP_real_real(tan,zero_zero_real) = zero_zero_real ).

tff(fact_3130_rel__simps_I2_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).

tff(fact_3131_rel__simps_I19_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).

tff(fact_3132_mult__Pls,axiom,
    ! [W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls ).

tff(fact_3133_add__Pls,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_2) = K_2 ).

tff(fact_3134_add__Pls__right,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),pls) = K_2 ).

tff(fact_3135_diff__bin__simps_I1_J,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls) = K_2 ).

tff(fact_3136_rel__simps_I40_J,axiom,
    min != pls ).

tff(fact_3137_rel__simps_I37_J,axiom,
    pls != min ).

tff(fact_3138_minus__Pls,axiom,
    hAPP_int_int(uminus_uminus_int,pls) = pls ).

tff(fact_3139_tan__minus,axiom,
    ! [X: real] : hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X)) ).

tff(fact_3140_tan__arctan,axiom,
    ! [Y: real] : hAPP_real_real(tan,hAPP_real_real(arctan,Y)) = Y ).

tff(fact_3141_semiring__norm_I113_J,axiom,
    zero_zero_nat = hAPP_int_nat(number_number_of_nat,pls) ).

tff(fact_3142_nat__number__of__Pls,axiom,
    hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).

tff(fact_3143_bin__less__0__simps_I1_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).

tff(fact_3144_zero__is__num__zero,axiom,
    zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).

tff(fact_3145_rel__simps_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).

tff(fact_3146_rel__simps_I3_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).

tff(fact_3147_tan__pi,axiom,
    hAPP_real_real(tan,pi) = zero_zero_real ).

tff(fact_3148_rel__simps_I23_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).

tff(fact_3149_rel__simps_I20_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).

tff(fact_3150_eq__number__of__Pls__Min,axiom,
    hAPP_int_int(number_number_of_int,pls) != hAPP_int_int(number_number_of_int,min) ).

tff(fact_3151_tan__periodic__pi,axiom,
    ! [X: real] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(tan,X) ).

tff(fact_3152_not__neg__number__of__Pls,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,pls))) ).

tff(fact_3153_succ__Min,axiom,
    hAPP_int_int(succ,min) = pls ).

tff(fact_3154_eq__0__number__of,axiom,
    ! [V: int] :
      ( ( zero_zero_nat = hAPP_int_nat(number_number_of_nat,V) )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ).

tff(fact_3155_eq__number__of__0,axiom,
    ! [V: int] :
      ( ( hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ).

tff(fact_3156_less__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1)) )
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) ) ) ).

tff(fact_3157_number__of2,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,pls))) ).

tff(fact_3158_le__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)))
    <=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),V_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ) ).

tff(fact_3159_less__0__number__of,axiom,
    ! [V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)) ) ).

tff(fact_3160_add__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_3) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3)) ) ) ) ) ) ).

tff(fact_3161_tan__npi,axiom,
    ! [N_1: nat] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = zero_zero_real ).

tff(fact_3162_tan__periodic__n,axiom,
    ! [X: real,N_1: int] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N_1)),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3163_tan__periodic__nat,axiom,
    ! [X: real,N_1: nat] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3164_tan__periodic__int,axiom,
    ! [X: real,I: int] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,I)),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3165_nat__number__of__mult__left,axiom,
    ! [V_3: int,K_2: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3))),K_2) ) ) ) ).

tff(fact_3166_mult__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3)) ) ) ) ).

tff(fact_3167_nat__number__of__diff__1,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2)) ) ) ) ).

tff(fact_3168_Suc__diff__number__of,axiom,
    ! [M_1: nat,V_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2))) ) ) ).

tff(fact_3169_rel__simps_I51_J,axiom,
    ! [K_1: int,L: int] :
      ( ( hAPP_int_int(bit1,K_1) = hAPP_int_int(bit1,L) )
    <=> ( K_1 = L ) ) ).

tff(fact_3170_minus__Bit1,axiom,
    ! [K_2: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,hAPP_int_int(uminus_uminus_int,K_2))) ).

tff(fact_3171_rel__simps_I46_J,axiom,
    ! [K_2: int] : hAPP_int_int(bit1,K_2) != pls ).

tff(fact_3172_rel__simps_I39_J,axiom,
    ! [L_2: int] : pls != hAPP_int_int(bit1,L_2) ).

tff(fact_3173_rel__simps_I17_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_3174_less__int__code_I16_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_3175_rel__simps_I34_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_3176_less__eq__int__code_I16_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_3177_rel__simps_I47_J,axiom,
    ! [K_1: int] :
      ( ( hAPP_int_int(bit1,K_1) = min )
    <=> ( K_1 = min ) ) ).

tff(fact_3178_rel__simps_I43_J,axiom,
    ! [L: int] :
      ( ( min = hAPP_int_int(bit1,L) )
    <=> ( min = L ) ) ).

tff(fact_3179_Bit1__Min,axiom,
    hAPP_int_int(bit1,min) = min ).

tff(fact_3180_succ__pred,axiom,
    ! [X: int] : hAPP_int_int(succ,hAPP_int_int(pred,X)) = X ).

tff(fact_3181_rel__simps_I12_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).

tff(fact_3182_bin__less__0__simps_I4_J,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,W_1)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).

tff(fact_3183_rel__simps_I22_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).

tff(fact_3184_Bit1__def,axiom,
    ! [K_2: int] : hAPP_int_int(bit1,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_2)),K_2) ).

tff(fact_3185_rel__simps_I13_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),min)) ) ).

tff(fact_3186_rel__simps_I9_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).

tff(fact_3187_rel__simps_I30_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).

tff(fact_3188_rel__simps_I26_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K_1)) ) ).

tff(fact_3189_neg__number__of__Bit1,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_1))))
    <=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) ) ).

tff(fact_3190_succ__Pls,axiom,
    hAPP_int_int(succ,pls) = hAPP_int_int(bit1,pls) ).

tff(fact_3191_le__iff__pred__less,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K_1)),L)) ) ).

tff(fact_3192_pred__Pls,axiom,
    hAPP_int_int(pred,pls) = min ).

tff(fact_3193_pred__def,axiom,
    ! [K_2: int] : hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),one_one_int) ).

tff(fact_3194_add__Min,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,min),K_2) = hAPP_int_int(pred,K_2) ).

tff(fact_3195_add__Min__right,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),min) = hAPP_int_int(pred,K_2) ).

tff(fact_3196_rel__simps_I5_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).

tff(fact_3197_rel__simps_I29_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).

tff(fact_3198_one__is__num__one,axiom,
    one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).

tff(fact_3199_Numeral1__eq1__nat,axiom,
    one_one_nat = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) ).

tff(fact_3200_nat__numeral__1__eq__1,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).

tff(fact_3201_diff__bin__simps_I4_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).

tff(fact_3202_minus__Min,axiom,
    hAPP_int_int(uminus_uminus_int,min) = hAPP_int_int(bit1,pls) ).

tff(fact_3203_numeral__3__eq__3,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_3204_numeral__1__eq__Suc__0,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3205_Suc3__eq__add__3,axiom,
    ! [N_1: nat] : hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1) ).

tff(fact_3206_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3207_transfer__nat__int__numerals_I4_J,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3208_power3__minus,axiom,
    ! [A_2: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3209_abs__power3__distrib,axiom,
    ! [X: int] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3210_Suc__diff__eq__diff__pred,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,pls)),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3211_Suc__div__eq__add3__div__number__of,axiom,
    ! [M_1: nat,V_2: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M_1)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M_1)),hAPP_int_nat(number_number_of_nat,V_2)) ).

tff(fact_3212_Suc__div__eq__add3__div,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M_1)))),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M_1)),N_1) ).

tff(fact_3213_div__Suc__eq__div__add3,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1)) ).

tff(fact_3214_Suc__mod__eq__add3__mod__number__of,axiom,
    ! [M_1: nat,V_2: int] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M_1)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M_1)),hAPP_int_nat(number_number_of_nat,V_2)) ).

tff(fact_3215_Suc__mod__eq__add3__mod,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M_1)))),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M_1)),N_1) ).

tff(fact_3216_mod__Suc__eq__mod__add3,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(div_mod_nat(M_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)))) = hAPP_nat_nat(div_mod_nat(M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N_1)) ).

tff(fact_3217_neg__number__of__pred__iff__0,axiom,
    ! [V: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
    <=> ( hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat ) ) ).

tff(fact_3218_tan__double,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) != zero_zero_real )
       => ( hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(tan,X))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3219_cos__pi__eq__zero,axiom,
    ! [M_1: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3220_Euler_Oaux____2,axiom,
    ! [P_3: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3221_neg__zmod__mult__2,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),one_one_int)),A_2))),one_one_int) ) ) ).

tff(fact_3222_zmod__number__of__Bit1,axiom,
    ! [V_2: int,W: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)))),one_one_int) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)))),one_one_int) ) ) ) ).

tff(fact_3223_arctan,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y)))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
      & ( hAPP_real_real(tan,hAPP_real_real(arctan,Y)) = Y ) ) ).

tff(fact_3224_rel__simps_I48_J,axiom,
    ! [K_1: int,L: int] :
      ( ( hAPP_int_int(bit0,K_1) = hAPP_int_int(bit0,L) )
    <=> ( K_1 = L ) ) ).

tff(fact_3225_rel__simps_I49_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(bit0,K_2) != hAPP_int_int(bit1,L_2) ).

tff(fact_3226_rel__simps_I50_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(bit1,K_2) != hAPP_int_int(bit0,L_2) ).

tff(fact_3227_Bit0__Pls,axiom,
    hAPP_int_int(bit0,pls) = pls ).

tff(fact_3228_rel__simps_I38_J,axiom,
    ! [L: int] :
      ( ( pls = hAPP_int_int(bit0,L) )
    <=> ( pls = L ) ) ).

tff(fact_3229_rel__simps_I44_J,axiom,
    ! [K_1: int] :
      ( ( hAPP_int_int(bit0,K_1) = pls )
    <=> ( K_1 = pls ) ) ).

tff(fact_3230_less__int__code_I13_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_3231_rel__simps_I14_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_3232_less__eq__int__code_I13_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_3233_rel__simps_I31_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_3234_mult__Bit0,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit0,K_2)),L_2) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2)) ).

tff(fact_3235_Bit0__def,axiom,
    ! [K_2: int] : hAPP_int_int(bit0,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),K_2) ).

tff(fact_3236_add__Bit0__Bit0,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).

tff(fact_3237_diff__bin__simps_I7_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).

tff(fact_3238_rel__simps_I42_J,axiom,
    ! [L_2: int] : min != hAPP_int_int(bit0,L_2) ).

tff(fact_3239_rel__simps_I45_J,axiom,
    ! [K_2: int] : hAPP_int_int(bit0,K_2) != min ).

tff(fact_3240_minus__Bit0,axiom,
    ! [K_2: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit0,hAPP_int_int(uminus_uminus_int,K_2)) ).

tff(fact_3241_less__int__code_I15_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_3242_rel__simps_I16_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_3243_rel__simps_I10_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).

tff(fact_3244_rel__simps_I4_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_1)) ) ).

tff(fact_3245_bin__less__0__simps_I3_J,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,W_1)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).

tff(fact_3246_rel__simps_I32_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_3247_less__eq__int__code_I14_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_3248_rel__simps_I27_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),pls)) ) ).

tff(fact_3249_rel__simps_I21_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).

tff(fact_3250_add__Bit1__Bit0,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).

tff(fact_3251_add__Bit0__Bit1,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).

tff(fact_3252_diff__bin__simps_I9_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).

tff(fact_3253_diff__bin__simps_I10_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).

tff(fact_3254_diff__bin__simps_I3_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),L_2)) ).

tff(fact_3255_rel__simps_I8_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).

tff(fact_3256_rel__simps_I28_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).

tff(fact_3257_zdiv__number__of__Bit0,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) ).

tff(fact_3258_neg__number__of__Bit0,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_1))))
    <=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) ) ).

tff(fact_3259_pred__Bit1,axiom,
    ! [K_2: int] : hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2) ).

tff(fact_3260_pred__Bit0,axiom,
    ! [K_2: int] : hAPP_int_int(pred,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,K_2)) ).

tff(fact_3261_pred__Min,axiom,
    hAPP_int_int(pred,min) = hAPP_int_int(bit0,min) ).

tff(fact_3262_succ__Bit1,axiom,
    ! [K_2: int] : hAPP_int_int(succ,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,hAPP_int_int(succ,K_2)) ).

tff(fact_3263_succ__Bit0,axiom,
    ! [K_2: int] : hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2) ).

tff(fact_3264_rel__simps_I15_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_3265_less__int__code_I14_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_3266_rel__simps_I33_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_3267_less__eq__int__code_I15_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_3268_mult__Bit1,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit1,K_2)),L_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))),L_2) ).

tff(fact_3269_rel__simps_I11_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).

tff(fact_3270_rel__simps_I25_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).

tff(fact_3271_diff__bin__simps_I6_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).

tff(fact_3272_diff__bin__simps_I5_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).

tff(fact_3273_diff__bin__simps_I8_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(pred,K_2)),L_2)) ).

tff(fact_3274_add__Bit1__Bit1,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),hAPP_int_int(succ,L_2))) ).

tff(fact_3275_number__of1,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N_1)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,N_1))))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,N_1)))) ) ) ).

tff(fact_3276_quartic__square__square,axiom,
    ! [X: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3277_numeral__2__eq__2,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_3278_semiring__norm_I115_J,axiom,
    hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_3279_pos2,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3280_add__2__eq__Suc_H,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)) ).

tff(fact_3281_add__2__eq__Suc,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N_1)) ).

tff(fact_3282_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3283_nat__mult__2,axiom,
    ! [Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) ).

tff(fact_3284_nat__mult__2__right,axiom,
    ! [Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) ).

tff(fact_3285_nat__1__add__1,axiom,
    hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_3286_transfer__nat__int__numerals_I3_J,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3287_eq__divide__2__times__iff,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_2)) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),Z_2) ) ) ).

tff(fact_3288_cos__two__neq__zero,axiom,
    hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_real ).

tff(fact_3289_div2__Suc__Suc,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3290_power2__ge__self,axiom,
    ! [X: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3291_mod2__Suc__Suc,axiom,
    ! [M_1: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,M_1))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(div_mod_nat(M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3292_pi__ge__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) ).

tff(fact_3293_pi__less__4,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3294_cube__square,axiom,
    ! [A_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3295_real__sum__of__halves,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X ).

tff(fact_3296_add__self__div__2,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),M_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = M_1 ).

tff(fact_3297_zmod__number__of__Bit0,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) ).

tff(fact_3298_pi__half__neq__two,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_3299_power2__eq__iff__abs__eq,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Aa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Ba),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
    <=> ( hAPP_int_int(abs_abs_int,Aa) = hAPP_int_int(abs_abs_int,Ba) ) ) ).

tff(fact_3300_abs__power2__distrib,axiom,
    ! [A_2: int] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3301_zEvenI,axiom,
    ! [X_1: int,K_1: int] :
      ( ( X_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_3302_less__2__cases,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
     => ( ( N_1 = zero_zero_nat )
        | ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_3303_nat__2,axiom,
    hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_3304_realpow__two__sum__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_3305_cos__two__le__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).

tff(fact_3306_cos__two__less__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).

tff(fact_3307_four__x__squared,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3308_two__realpow__ge__one,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) ).

tff(fact_3309_neq__one__mod__two,axiom,
    ! [X_1: int] :
      ( ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int )
    <=> ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int ) ) ).

tff(fact_3310_two__realpow__gt,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) ).

tff(fact_3311_real__gt__half__sum,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)) ) ).

tff(fact_3312_real__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3313_realpow__square__minus__le,axiom,
    ! [U_1: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,U_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3314_mod__exhaust__less__4,axiom,
    ! [M_1: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_nat )
      | ( hAPP_nat_nat(div_mod_nat(M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat )
      | ( hAPP_nat_nat(div_mod_nat(M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) )
      | ( hAPP_nat_nat(div_mod_nat(M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) ) ) ).

tff(fact_3315_pi__half__le__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3316_pi__half__neq__zero,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_real ).

tff(fact_3317_pi__half__less__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3318_real__average__minus__second,axiom,
    ! [B_1: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),A_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3319_real__average__minus__first,axiom,
    ! [A_2: real,B_1: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_2),B_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3320_power2__eq1__iff,axiom,
    ! [A_2: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int )
     => ( hAPP_int_int(abs_abs_int,A_2) = one_one_int ) ) ).

tff(fact_3321_even__div__2__prop1,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ) ).

tff(fact_3322_even__div__2__l,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3323_even__div__2__prop2,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1 ) ) ).

tff(fact_3324_even__prod__div__2,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_2) ) ) ).

tff(fact_3325_even__sum__div__2,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ).

tff(fact_3326_int__pos__lt__two__imp__zero__or__one,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => ( ( X = zero_zero_int )
          | ( X = one_one_int ) ) ) ) ).

tff(fact_3327_Suc__Suc__mult__two__diff__two,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) ) ) ).

tff(fact_3328_lemma__Suc__Suc__4n__diff__2,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1) ) ) ).

tff(fact_3329_div__2__gt__zero,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3330_mod2__gr__0,axiom,
    ! [M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
    <=> ( hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_nat ) ) ).

tff(fact_3331_fact__lemma,axiom,
    ! [N_1: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)) ).

tff(fact_3332_pi__half__ge__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3333_pi__half__gt__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3334_zdiv__number__of__Bit1,axiom,
    ! [V_2: int,W: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)) ) ) ) ).

tff(fact_3335_less__one__imp__sqr__less__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_real)) ) ).

tff(fact_3336_zadd__power2,axiom,
    ! [A_2: int,B_1: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_2)),B_1))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3337_zadd__power3,axiom,
    ! [A_2: int,B_1: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3338_diff__square,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).

tff(fact_3339_zspecial__product,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3340_m2pi__less__pi,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))),pi)) ).

tff(fact_3341_cos__pi__half,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3342_cos__two__pi,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) = one_one_real ).

tff(fact_3343_cos__periodic,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(cos,X) ).

tff(fact_3344_cos__60,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3345_zOddI,axiom,
    ! [X_1: int,K_1: int] :
      ( ( X_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1)),one_one_int) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) ) ).

tff(fact_3346_tan__periodic,axiom,
    ! [X: real] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3347_tan__45,axiom,
    hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = one_one_real ).

tff(fact_3348_arctan__ubound,axiom,
    ! [Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3349_arctan1__eq__pi4,axiom,
    hAPP_real_real(arctan,one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3350_Suc__mult__two__diff__one,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1) ) ) ).

tff(fact_3351_Int2_Oaux__2,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3352_Suc__n__div__2__gt__zero,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3353_cos__double__less__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),one_one_real)) ) ) ).

tff(fact_3354_Int2_Oaux__1,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,P_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ).

tff(fact_3355_cos__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).

tff(fact_3356_zdiff__power3,axiom,
    ! [A_2: int,B_1: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3357_zdiff__power2,axiom,
    ! [A_2: int,B_1: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_2)),B_1))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3358_minus__pi__half__less__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) ).

tff(fact_3359_cos__3over2__pi,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),pi)) = zero_zero_real ).

tff(fact_3360_cos__2npi,axiom,
    ! [N_1: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N_1))),pi)) = one_one_real ).

tff(fact_3361_tan__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(tan,X))) ) ) ).

tff(fact_3362_tan__monotone,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y)),hAPP_real_real(tan,X))) ) ) ) ).

tff(fact_3363_tan__monotone_H,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
            <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_2)),hAPP_real_real(tan,X_1))) ) ) ) ) ) ).

tff(fact_3364_tan__inverse,axiom,
    ! [Y: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y)) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)) ).

tff(fact_3365_arctan__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y)))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3366_arctan__lbound,axiom,
    ! [Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y))) ).

tff(fact_3367_neg__zdiv__mult__2,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),one_one_int)),A_2) ) ) ).

tff(fact_3368_pos__zdiv__mult__2,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),A_2) ) ) ).

tff(fact_3369_pos__zmod__mult__2,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(B_1),A_2))) ) ) ).

tff(fact_3370_cos__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).

tff(fact_3371_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).

tff(fact_3372_Euler_Oaux__2,axiom,
    ! [Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Pa),zOdd))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3373_tan__less__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X)),zero_zero_real)) ) ) ).

tff(fact_3374_machin,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))))))))) ).

tff(fact_3375_arctan__tan,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hAPP_real_real(arctan,hAPP_real_real(tan,X)) = X ) ) ) ).

tff(fact_3376_tan__total__pi4,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Z))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
          & ( hAPP_real_real(tan,Z) = X ) ) ) ).

tff(fact_3377_best__odd__division__abs,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
       => ? [N_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_1))))),X_1)) ) ) ).

tff(fact_3378_tan__total__pos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
          & ( hAPP_real_real(tan,X_2) = Y ) ) ) ).

tff(fact_3379_best__division__abs,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ? [N_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X))))),X)) ) ).

tff(fact_3380_inv__inv__aux,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))))) ) ) ).

tff(fact_3381_inv__is__inv__aux,axiom,
    ! [M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_1))
     => ( hAPP_nat_nat(suc,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M_1),one_one_int)) ) ) ).

tff(fact_3382_tan__total,axiom,
    ! [Y: real] :
    ? [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
      & ( hAPP_real_real(tan,X_2) = Y )
      & ! [Y_1: real] :
          ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
            & ( hAPP_real_real(tan,Y_1) = Y ) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_3383_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
      & ( hAPP_real_real(tan,X_2) = Y ) ) ).

tff(fact_3384_lemma__tan__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(tan,X_2))) ) ) ).

tff(fact_3385_zOddE,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ~ ! [K: int] : X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K)),one_one_int) ) ).

tff(fact_3386_real__sum__squared__expand,axiom,
    ! [X: real,Y: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),Y)) ).

tff(fact_3387_cos__is__zero,axiom,
    ? [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
      & ( hAPP_real_real(cos,X_2) = zero_zero_real )
      & ! [Y_1: real] :
          ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
            & ( hAPP_real_real(cos,Y_1) = zero_zero_real ) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_3388_zEvenE,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ~ ! [K: int] : X_1 != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K) ) ).

tff(fact_3389_negDivAlg__eqn__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))))
         => ( negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))))
         => ( negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),negDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) ) ) ) ) ).

tff(fact_3390_negDivAlg__eqn__number__of,axiom,
    ! [W: int,V_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))))
         => ( negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))))
         => ( negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) ) ) ) ) ).

tff(fact_3391_negDivAlg__eqn,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)))
         => ( negDivAlg(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)))
         => ( negDivAlg(A_2,B_1) = hAPP_P1975530577nt_int(adjust(B_1),negDivAlg(A_2,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))) ) ) ) ) ).

tff(fact_3392_posDivAlg__eqn,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1))
         => ( posDivAlg(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_2) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1))
         => ( posDivAlg(A_2,B_1) = hAPP_P1975530577nt_int(adjust(B_1),posDivAlg(A_2,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))) ) ) ) ) ).

tff(fact_3393_posDivAlg__eqn__number__of,axiom,
    ! [W: int,V_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))
         => ( posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))
         => ( posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) ) ) ) ) ).

tff(fact_3394_posDivAlg_Osimps,axiom,
    ! [A_2: int,B_1: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
       => ( posDivAlg(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_2) ) )
      & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_1))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
       => ( posDivAlg(A_2,B_1) = hAPP_P1975530577nt_int(adjust(B_1),posDivAlg(A_2,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))) ) ) ) ).

tff(fact_3395_posDivAlg__eqn__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W)))
         => ( posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W)))
         => ( posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),posDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) ) ) ) ) ).

tff(fact_3396_negDivAlg_Osimps,axiom,
    ! [A_2: int,B_1: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
       => ( negDivAlg(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)) ) )
      & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_1)))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
       => ( negDivAlg(A_2,B_1) = hAPP_P1975530577nt_int(adjust(B_1),negDivAlg(A_2,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_1))) ) ) ) ).

tff(fact_3397_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3398_arcsin__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3399_arcsin__lbound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) ) ) ).

tff(fact_3400_arcsin__ubound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3401_lemma__STAR__cos,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_1)) = zero_zero_real ) ) ).

tff(fact_3402_real__of__nat__fact__not__zero,axiom,
    ! [N_1: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)) != zero_zero_real ).

tff(fact_3403_real__of__nat__fact__ge__zero,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))) ).

tff(fact_3404_real__of__nat__fact__gt__zero,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))) ).

tff(fact_3405_fact__diff__Suc,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M_1)))
     => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1))) ) ) ).

tff(fact_3406_fact__add__num__eq__if2__nat,axiom,
    ! [N_1: nat,M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_nat_nat(fact_fact_nat,N_1) ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1))) ) ) ) ).

tff(fact_3407_fact__add__num__eq__if__nat,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = one_one_nat ) )
      & ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) != zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),one_one_nat))) ) ) ) ).

tff(fact_3408_fact__reduce__nat,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_nat_nat(fact_fact_nat,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))) ) ) ).

tff(fact_3409_fact__num__eq__if__nat,axiom,
    ! [M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,M_1) = one_one_nat ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat))) ) ) ) ).

tff(fact_3410_lemma__STAR__sin,axiom,
    ! [N: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,hAPP_nat_bool(even_odd_even_nat,N)),zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N)) = zero_zero_real ).

tff(fact_3411_odd__1__nat,axiom,
    ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,one_one_nat)) ).

tff(fact_3412_even__product__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_2)))
    <=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
        | hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ).

tff(fact_3413_even__sum__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) )
        | ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ) ).

tff(fact_3414_even__add,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,M))
      <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) ) ) ).

tff(fact_3415_odd__add,axiom,
    ! [M: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ~ ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,M))
        <=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) ) ) ).

tff(fact_3416_even__Suc,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,X_1)))
    <=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) ) ).

tff(fact_3417_even__zero__nat,axiom,
    hBOOL(hAPP_nat_bool(even_odd_even_nat,zero_zero_nat)) ).

tff(fact_3418_odd__pos,axiom,
    ! [N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3419_even__difference__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_2)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
        | ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) )
        | ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ) ).

tff(fact_3420_even__nat__equiv__def2,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ? [Y_1: nat] : X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1) ) ).

tff(fact_3421_odd__nat__equiv__def2,axiom,
    ! [X_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ? [Y_1: nat] : X_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1)) ) ).

tff(fact_3422_even__nat__plus__one__div__two,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ) ) ).

tff(fact_3423_odd__nat__plus__one__div__two,axiom,
    ! [X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) ) ) ).

tff(fact_3424_odd__nat__mod__two__eq__one,axiom,
    ! [X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

tff(fact_3425_even__nat__mod__two__eq__zero,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ) ).

tff(fact_3426_odd__nat__equiv__def,axiom,
    ! [X_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ( hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

tff(fact_3427_even__nat__equiv__def,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ( hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ) ).

tff(fact_3428_even__power__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),Y_2)))
    <=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_2)) ) ) ).

tff(fact_3429_even__num__iff,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
      <=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

tff(fact_3430_even__nat__div__two__times__two,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = X ) ) ).

tff(fact_3431_odd__nat__div__two__times__two__plus__one,axiom,
    ! [X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) = X ) ) ).

tff(fact_3432_even__mult__two__ex,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
    <=> ? [M_2: nat] : N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_2) ) ).

tff(fact_3433_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
    <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3434_odd__Suc__mult__two__ex,axiom,
    ! [N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
    <=> ? [M_2: nat] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_2)) ) ).

tff(fact_3435_lemma__even__div2,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3436_fact__nonzero__nat,axiom,
    ! [N_1: nat] : hAPP_nat_nat(fact_fact_nat,N_1) != zero_zero_nat ).

tff(fact_3437_fact__mono__nat,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact_fact_nat,M_1)),hAPP_nat_nat(fact_fact_nat,N_1))) ) ).

tff(fact_3438_fact__1__nat,axiom,
    hAPP_nat_nat(fact_fact_nat,one_one_nat) = one_one_nat ).

tff(fact_3439_lemma__not__even__div2,axiom,
    ! [N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ).

tff(fact_3440_lemma__odd__mod__4__div__2,axiom,
    ! [N_1: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) )
     => ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3441_lemma__even__mod__4__div__2,axiom,
    ! [N_1: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat )
     => hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3442_Fact_Ofact__0__nat,axiom,
    hAPP_nat_nat(fact_fact_nat,zero_zero_nat) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3443_fact__Suc__0__nat,axiom,
    hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3444_fact__less__mono__nat,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M_1)),hAPP_nat_nat(fact_fact_nat,N_1))) ) ) ).

tff(fact_3445_fact__gt__zero__nat,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N_1))) ).

tff(fact_3446_cos__zero__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) = zero_zero_real )
    <=> ( ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) )
        | ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ) ).

tff(fact_3447_Fact_Ofact__Suc,axiom,
    ! [X: nat] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(fact_fact_nat,X)) ).

tff(fact_3448_fact__ge__one__nat,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N_1))) ).

tff(fact_3449_fact__ge__Suc__0__nat,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(fact_fact_nat,N_1))) ).

tff(fact_3450_fact__plus__one__nat,axiom,
    ! [N_1: nat] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N_1)) ).

tff(fact_3451_fact__mod,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N_1)),hAPP_nat_nat(fact_fact_nat,M_1)) = zero_zero_nat ) ) ).

tff(fact_3452_cos__zero__lemma,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( hAPP_real_real(cos,X) = zero_zero_real )
       => ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ) ).

tff(fact_3453_odd__square,axiom,
    ! [N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ? [X_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2)),one_one_nat) ) ).

tff(fact_3454_sin__coeff__def,axiom,
    ! [X_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(sin_coeff,X_2) = zero_zero_real ) )
      & ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(sin_coeff,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2))) ) ) ) ).

tff(fact_3455_even__square,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ? [X_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2) ) ).

tff(fact_3456_cos__coeff__def,axiom,
    ! [X_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(cos_coeff,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2))) ) )
      & ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(cos_coeff,X_2) = zero_zero_real ) ) ) ).

tff(fact_3457_WilsonRuss_Oinv__def,axiom,
    ! [P_3: int,A_2: int] : hAPP_int_int(inv(P_3),A_2) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3) ).

tff(fact_3458_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sgn_sgn_real,X)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X)) ) ) ).

tff(fact_3459_tan__half,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hAPP_real_real(tan,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),one_one_real)) ) ) ) ).

tff(fact_3460_sin__cos__npi,axiom,
    ! [N_1: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N_1) ).

tff(fact_3461_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).

tff(fact_3462_sin__zero,axiom,
    hAPP_real_real(sin,zero_zero_real) = zero_zero_real ).

tff(fact_3463_sin__minus,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).

tff(fact_3464_sin__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X)),one_one_real)) ).

tff(fact_3465_sin__pi,axiom,
    hAPP_real_real(sin,pi) = zero_zero_real ).

tff(fact_3466_ln__less__self,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),X)) ) ).

tff(fact_3467_ln__inj__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
       => ( ( hAPP_real_real(ln,X_1) = hAPP_real_real(ln,Y_2) )
        <=> ( X_1 = Y_2 ) ) ) ) ).

tff(fact_3468_ln__less__cancel__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_2)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ) ).

tff(fact_3469_ln__one,axiom,
    hAPP_real_real(ln,one_one_real) = zero_zero_real ).

tff(fact_3470_real__sgn__eq,axiom,
    ! [X: real] : hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(abs_abs_real,X)) ).

tff(fact_3471_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( ( hAPP_real_real(ln,X) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

tff(fact_3472_sin__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3473_sin__gt__zero__pi,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),pi))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3474_sin__ge__minus__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(sin,X))) ).

tff(fact_3475_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(cos,X) = one_one_real )
     => ( hAPP_real_real(sin,X) = zero_zero_real ) ) ).

tff(fact_3476_abs__sin__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X))),one_one_real)) ).

tff(fact_3477_sin__add,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3478_sin__diff,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3479_sin__diff2,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(cos,X))) ).

tff(fact_3480_sin__periodic__pi2,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).

tff(fact_3481_sin__periodic__pi,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).

tff(fact_3482_ln__le__cancel__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_2)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ) ).

tff(fact_3483_tan__def,axiom,
    ! [X: real] : hAPP_real_real(tan,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,X)) ).

tff(fact_3484_ln__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X))) ) ).

tff(fact_3485_ln__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X))) ) ).

tff(fact_3486_ln__gt__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) ) ) ).

tff(fact_3487_ln__eq__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( ( hAPP_real_real(ln,X_1) = zero_zero_real )
      <=> ( X_1 = one_one_real ) ) ) ).

tff(fact_3488_ln__less__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ) ).

tff(fact_3489_ln__less__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),zero_zero_real)) ) ) ).

tff(fact_3490_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) ) ) ).

tff(fact_3491_real__sgn__pos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(sgn_sgn_real,X) = one_one_real ) ) ).

tff(fact_3492_ln__le__minus__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real))) ) ).

tff(fact_3493_sin__npi2,axiom,
    ! [N_1: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N_1))) = zero_zero_real ).

tff(fact_3494_sin__npi,axiom,
    ! [N_1: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),pi)) = zero_zero_real ).

tff(fact_3495_sin__cos__squared__add3,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,X))) = one_one_real ).

tff(fact_3496_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(sin,X) = zero_zero_real )
     => ( hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X)) = one_one_real ) ) ).

tff(fact_3497_cos__diff2,axiom,
    ! [X: real,Y: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) ).

tff(fact_3498_cos__diff,axiom,
    ! [X: real,Y: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3499_cos__add,axiom,
    ! [X: real,Y: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3500_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

tff(fact_3501_ln__ge__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ) ).

tff(fact_3502_ln__mult,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) ) ) ) ).

tff(fact_3503_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)) ) ).

tff(fact_3504_ln__div,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) ) ) ) ).

tff(fact_3505_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)) ) ).

tff(fact_3506_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))),hAPP_real_real(uminus_uminus_real,X))) ) ) ).

tff(fact_3507_aux5,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
     => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))))) ) ) ).

tff(fact_3508_real__sgn__neg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
     => ( hAPP_real_real(sgn_sgn_real,X) = hAPP_int_real(number267125858f_real,min) ) ) ).

tff(fact_3509_sgn__real__def,axiom,
    ! [A_2: real] :
      ( ( ( A_2 = zero_zero_real )
       => ( hAPP_real_real(sgn_sgn_real,A_2) = zero_zero_real ) )
      & ( ( A_2 != zero_zero_real )
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
           => ( hAPP_real_real(sgn_sgn_real,A_2) = one_one_real ) )
          & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
           => ( hAPP_real_real(sgn_sgn_real,A_2) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

tff(fact_3510_real__sgn__def,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( hAPP_real_real(sgn_sgn_real,X) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
           => ( hAPP_real_real(sgn_sgn_real,X) = one_one_real ) )
          & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
           => ( hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

tff(fact_3511_sin__arcsin,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ).

tff(fact_3512_ln__realpow,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(ln,X)) ) ) ).

tff(fact_3513_sin__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3514_sin__double,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(sin,X))),hAPP_real_real(cos,X)) ).

tff(fact_3515_add__tan__eq,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,Y) != zero_zero_real )
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3516_sin__two__pi,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) = zero_zero_real ).

tff(fact_3517_sin__periodic,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(sin,X) ).

tff(fact_3518_sin__30,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3519_sin__pi__half,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).

tff(fact_3520_sin__gt__zero2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3521_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)))) ) ) ).

tff(fact_3522_sin__cos__squared__add2__mult,axiom,
    ! [R_3: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_2))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_2))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3523_sin__2npi,axiom,
    ! [N_1: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N_1))),pi)) = zero_zero_real ).

tff(fact_3524_sin__cos__squared__add,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).

tff(fact_3525_sin__cos__squared__add2,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).

tff(fact_3526_sin__squared__eq,axiom,
    ! [X: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3527_cos__squared__eq,axiom,
    ! [X: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3528_sin__monotone__2pi_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) ) ) ) ).

tff(fact_3529_cos__sin__eq,axiom,
    ! [X: real] : hAPP_real_real(cos,X) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) ).

tff(fact_3530_sin__cos__eq,axiom,
    ! [X: real] : hAPP_real_real(sin,X) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) ).

tff(fact_3531_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3532_sin__less__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sin,X)),zero_zero_real)) ) ) ).

tff(fact_3533_cos__double,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3534_sin__3over2__pi,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),pi)) = hAPP_real_real(uminus_uminus_real,one_one_real) ).

tff(fact_3535_minus__sin__cos__eq,axiom,
    ! [X: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3536_arcsin__sin,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hAPP_real_real(arcsin,hAPP_real_real(sin,X)) = X ) ) ) ).

tff(fact_3537_sin__cos__add,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3538_sin__cos__minus,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(sin,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3539_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X)))) ) ) ).

tff(fact_3540_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3541_sin__expansion__lemma,axiom,
    ! [X: real,M_1: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M_1))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M_1)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3542_sin__zero__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(sin,X_1) = zero_zero_real )
    <=> ( ? [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) )
        | ? [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ) ).

tff(fact_3543_arcsin__pi,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),pi))
          & ( hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ) ).

tff(fact_3544_arcsin,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
          & ( hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ) ).

tff(fact_3545_cos__expansion__lemma,axiom,
    ! [X: real,M_1: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M_1))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M_1)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_3546_sin__zero__lemma,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( hAPP_real_real(sin,X) = zero_zero_real )
       => ? [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ) ).

tff(fact_3547_sin__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
            & ( hAPP_real_real(sin,X_2) = Y )
            & ! [Y_1: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
                  & ( hAPP_real_real(sin,Y_1) = Y ) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_3548_Ln_Oaux1,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N_1)))) ) ) ).

tff(fact_3549_cos__arcsin,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => ( hAPP_real_real(cos,hAPP_real_real(arcsin,X)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3550_sin__arccos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => ( hAPP_real_real(sin,hAPP_real_real(arccos,X)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3551_INVERSE__ZERO,axiom,
    hAPP_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

tff(fact_3552_real__sqrt__minus,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X)) ).

tff(fact_3553_real__sqrt__le__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ).

tff(fact_3554_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))) ) ).

tff(fact_3555_real__sqrt__eq__0__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(sqrt,X_1) = zero_zero_real )
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_3556_real__sqrt__zero,axiom,
    hAPP_real_real(sqrt,zero_zero_real) = zero_zero_real ).

tff(fact_3557_real__sqrt__inverse,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) ).

tff(fact_3558_real__sqrt__eq__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(sqrt,X_1) = hAPP_real_real(sqrt,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3559_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) ).

tff(fact_3560_real__sqrt__one,axiom,
    hAPP_real_real(sqrt,one_one_real) = one_one_real ).

tff(fact_3561_real__sqrt__eq__1__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(sqrt,X_1) = one_one_real )
    <=> ( X_1 = one_one_real ) ) ).

tff(fact_3562_real__sqrt__power,axiom,
    ! [X: real,K_2: nat] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),K_2) ).

tff(fact_3563_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) ).

tff(fact_3564_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))) ) ).

tff(fact_3565_real__sqrt__less__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ).

tff(fact_3566_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),X) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) ) ) ).

tff(fact_3567_real__sqrt__ge__0__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) ) ).

tff(fact_3568_real__sqrt__le__0__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).

tff(fact_3569_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,X))) ) ).

tff(fact_3570_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( hAPP_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

tff(fact_3571_real__sqrt__gt__0__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) ) ).

tff(fact_3572_real__sqrt__lt__0__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ).

tff(fact_3573_real__sqrt__not__eq__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(sqrt,X) != zero_zero_real ) ) ).

tff(fact_3574_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,X))) ) ).

tff(fact_3575_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,X))) ) ).

tff(fact_3576_real__sqrt__le__1__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).

tff(fact_3577_real__sqrt__ge__1__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) ) ).

tff(fact_3578_real__sqrt__gt__1__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) ) ).

tff(fact_3579_real__sqrt__lt__1__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ).

tff(fact_3580_real__sqrt__abs2,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)) = hAPP_real_real(abs_abs_real,X) ).

tff(fact_3581_real__divide__def,axiom,
    ! [R: real,S_5: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R),S_5) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R),hAPP_real_real(inverse_inverse_real,S_5)) ).

tff(fact_3582_divide__real__def,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),hAPP_real_real(inverse_inverse_real,Y)) ).

tff(fact_3583_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y))))) ).

tff(fact_3584_real__mult__inverse__cancel2,axiom,
    ! [Y: real,U_1: real,X1_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_2))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_2),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(inverse_inverse_real,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_2)))) ) ) ) ).

tff(fact_3585_real__mult__inverse__cancel,axiom,
    ! [Y: real,U_1: real,X1_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_2))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_2),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_2)),U_1))) ) ) ) ).

tff(fact_3586_real__mult__inverse__left,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),X) = one_one_real ) ) ).

tff(fact_3587_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(inverse_inverse_real,X) ) ) ).

tff(fact_3588_real__sqrt__mult__self__sum__ge__zero,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y))))) ).

tff(fact_3589_inv__real__of__nat__fact__ge__zero,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) ).

tff(fact_3590_inv__real__of__nat__fact__gt__zero,axiom,
    ! [N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) ).

tff(fact_3591_ln__inverse,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X)) ) ) ).

tff(fact_3592_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( hAPP_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I))
           => ( hAPP_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I))
           => ( hAPP_int_int(sgn_sgn_int,I) = hAPP_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

tff(fact_3593_real__sqrt__two__ge__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3594_real__sqrt__two__gt__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3595_real__sqrt__two__gt__one,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3596_real__sqrt__abs,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(abs_abs_real,X) ).

tff(fact_3597_tan__60,axiom,
    hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3598_real__sqrt__pow2__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_1 )
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) ) ).

tff(fact_3599_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X ) ) ).

tff(fact_3600_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( hAPP_real_real(sqrt,X) = Y ) ) ) ).

tff(fact_3601_real__sqrt__pow2__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3602_lemma__real__divide__sqrt__less,axiom,
    ! [U_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),U_1)) ) ).

tff(fact_3603_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A_2: real,C_1: real,B_1: real,D_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_2),C_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),D_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,C_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,D_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).

tff(fact_3604_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3605_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3606_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X )
     => ( Y = zero_zero_real ) ) ).

tff(fact_3607_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = Y )
     => ( X = zero_zero_real ) ) ).

tff(fact_3608_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa_2: real,Ya: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3609_cos__45,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3610_cos__30,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3611_sin__60,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3612_sin__45,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3613_real__sqrt__sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3614_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3615_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3616_tan__30,axiom,
    hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3617_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa_2: real,Ya: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).

tff(fact_3618_cos__arccos__lemma1,axiom,
    ! [X_4: real,Y_4: real] : hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_3619_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),one_one_real)) ).

tff(fact_3620_real__sqrt__sum__squares__less,axiom,
    ! [Y: real,X: real,U_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ).

tff(fact_3621_tan__sec,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3622_lemma__sqrt__hcomplex__capprox,axiom,
    ! [Y: real,X: real,U_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ) ) ) ).

tff(fact_3623_arctan__half,axiom,
    ! [X: real] : hAPP_real_real(arctan,X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ).

tff(fact_3624_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
     => ( hAPP_real_real(sin,hAPP_real_real(arccos,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3625_sin__arccos__lemma1,axiom,
    ! [X_4: real,Y_4: real] : hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3626_reals__Archimedean,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))),X)) ) ).

tff(fact_3627_DERIV__arccos,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(deriv_real(arccos,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ) ) ).

tff(fact_3628_DERIV__arcsin,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(deriv_real(arcsin,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ) ).

tff(fact_3629_Bolzano__bisect__diff,axiom,
    ! [P_1: fun_Pr1127657839l_bool,N: nat,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)) ) ) ).

tff(fact_3630_Bolzano__bisect_Osimps_I1_J,axiom,
    ! [P_1: fun_Pr1127657839l_bool,Aa: real,Ba: real] : hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),zero_zero_nat) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba) ).

tff(fact_3631_Bolzano__bisect__le,axiom,
    ! [P_1: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2)))) ) ).

tff(fact_3632_DERIV__sin,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(sin,X_1),hAPP_real_real(cos,X_1))) ).

tff(fact_3633_DERIV__cos,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(cos,X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)))) ).

tff(fact_3634_Bolzano__bisect__fst__le__Suc,axiom,
    ! [P_1: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),hAPP_nat_nat(suc,N_2))))) ) ).

tff(fact_3635_Bolzano__bisect__Suc__le__snd,axiom,
    ! [P_1: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),hAPP_nat_nat(suc,N_2)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2)))) ) ).

tff(fact_3636_DERIV__ln,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(ln,X_1),hAPP_real_real(inverse_inverse_real,X_1))) ) ).

tff(fact_3637_DERIV__ln__divide,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(ln,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_1))) ) ).

tff(fact_3638_DERIV__real__sqrt,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(sqrt,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3639_DERIV__tan,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) != zero_zero_real )
     => hBOOL(hAPP_real_bool(deriv_real(tan,X_1),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3640_DERIV__arctan,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(arctan,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3641_DERIV__real__sqrt__generic,axiom,
    ! [D_1: real,X_1: real] :
      ( ( X_1 != zero_zero_real )
     => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
         => ( D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) )
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
           => ( D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) )
         => hBOOL(hAPP_real_bool(deriv_real(sqrt,X_1),D_1)) ) ) ) ).

tff(fact_3642_DERIV__const__average,axiom,
    ! [V: fun_real_real,K_1: real,Aa: real,Ba: real] :
      ( ( Aa != Ba )
     => ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(V,X_2),K_1))
       => ( hAPP_real_real(V,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Aa),Ba)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(V,Aa)),hAPP_real_real(V,Ba))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ) ).

tff(fact_3643_DERIV__local__max,axiom,
    ! [D: real,F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y_1)),hAPP_real_real(F,X_1))) )
         => ( L = zero_zero_real ) ) ) ) ).

tff(fact_3644_DERIV__local__min,axiom,
    ! [D: real,F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,Y_1))) )
         => ( L = zero_zero_real ) ) ) ) ).

tff(fact_3645_DERIV__local__const,axiom,
    ! [D: real,F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D))
             => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_1) ) )
         => ( L = zero_zero_real ) ) ) ) ).

tff(fact_3646_not__P__Bolzano__bisect,axiom,
    ! [N: nat,Aa: real,Ba: real,P_1: fun_Pr1127657839l_bool] :
      ( ! [A_1: real,B_3: real,C_3: real] :
          ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3)))
         => ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_3),C_3)))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_3))
             => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),C_3))
               => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),C_3))) ) ) ) )
     => ( ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba)))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
         => ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N))))) ) ) ) ).

tff(fact_3647_not__P__Bolzano__bisect_H,axiom,
    ! [Aa: real,Ba: real,P_1: fun_Pr1127657839l_bool] :
      ( ! [A_1: real,B_3: real,C_3: real] :
          ( ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3)))
            & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_3),C_3)))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_3))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),C_3)) )
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),C_3))) )
     => ( ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba)))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
         => ! [N_2: nat] : ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2))))) ) ) ) ).

tff(fact_3648_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),H_1))),hAPP_real_real(F,X_1))) ) ) ) ) ) ).

tff(fact_3649_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),H_1)))) ) ) ) ) ) ).

tff(fact_3650_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),H_1)))) ) ) ) ) ) ).

tff(fact_3651_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),H_1))),hAPP_real_real(F,X_1))) ) ) ) ) ) ).

tff(fact_3652_DERIV__const__ratio__const2,axiom,
    ! [F: fun_real_real,K_1: real,Aa: real,Ba: real] :
      ( ( Aa != Ba )
     => ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(F,X_2),K_1))
       => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)) = K_1 ) ) ) ).

tff(fact_3653_DERIV__const__ratio__const,axiom,
    ! [F: fun_real_real,K_1: real,Aa: real,Ba: real] :
      ( ( Aa != Ba )
     => ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(F,X_2),K_1))
       => ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)),K_1) ) ) ) ).

tff(fact_3654_DERIV__real__root__generic,axiom,
    ! [D_1: real,X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X_1 != zero_zero_real )
       => ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
             => ( D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
             => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
               => ( D_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
               => ( D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) ) )
             => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_1),D_1)) ) ) ) ) ) ).

tff(fact_3655_DERIV__even__real__root,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ) ).

tff(fact_3656_ln__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),X)) ) ).

tff(fact_3657_DERIV__real__root,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).

tff(fact_3658_real__root__zero,axiom,
    ! [N_1: nat] : hAPP_real_real(hAPP_n546711566l_real(root,N_1),zero_zero_real) = zero_zero_real ).

tff(fact_3659_real__root__Suc__0,axiom,
    ! [X: real] : hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X) = X ).

tff(fact_3660_real__root__commute,axiom,
    ! [X: real,N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,M_1),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_n546711566l_real(root,M_1),X)) ) ) ) ).

tff(fact_3661_real__root__eq__iff,axiom,
    ! [X_1: real,Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_1) = hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2) )
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_3662_odd__real__root__power__cancel,axiom,
    ! [X: real,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1)) = X ) ) ).

tff(fact_3663_odd__real__root__pow,axiom,
    ! [X: real,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),N_1) = X ) ) ).

tff(fact_3664_odd__real__root__unique,axiom,
    ! [Y: real,X: real,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N_1) = X )
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X) = Y ) ) ) ).

tff(fact_3665_real__root__le__mono,axiom,
    ! [X: real,Y: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y))) ) ) ).

tff(fact_3666_real__root__le__iff,axiom,
    ! [X_1: real,Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ).

tff(fact_3667_real__root__eq__0__iff,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_1) = zero_zero_real )
      <=> ( X_1 = zero_zero_real ) ) ) ).

tff(fact_3668_real__root__less__mono,axiom,
    ! [X: real,Y: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y))) ) ) ).

tff(fact_3669_real__root__less__iff,axiom,
    ! [X_1: real,Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ).

tff(fact_3670_real__root__mult,axiom,
    ! [X: real,Y: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y)) ) ) ).

tff(fact_3671_real__root__mult__exp,axiom,
    ! [X: real,N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M_1),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) ) ) ) ).

tff(fact_3672_real__root__power,axiom,
    ! [X: real,K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),K_2) ) ) ).

tff(fact_3673_real__root__eq__1__iff,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_1) = one_one_real )
      <=> ( X_1 = one_one_real ) ) ) ).

tff(fact_3674_real__root__one,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),one_one_real) = one_one_real ) ) ).

tff(fact_3675_real__root__divide,axiom,
    ! [X: real,Y: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y)) ) ) ).

tff(fact_3676_real__root__minus,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) ) ) ).

tff(fact_3677_real__root__abs,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(abs_abs_real,X)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) ) ) ).

tff(fact_3678_real__root__inverse,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) ) ) ).

tff(fact_3679_real__root__ge__zero,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X))) ) ) ).

tff(fact_3680_real__root__le__0__iff,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ) ).

tff(fact_3681_real__root__ge__0__iff,axiom,
    ! [Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) ) ) ).

tff(fact_3682_real__root__gt__0__iff,axiom,
    ! [Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) ) ) ).

tff(fact_3683_real__root__lt__0__iff,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ) ).

tff(fact_3684_real__root__gt__zero,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X))) ) ) ).

tff(fact_3685_real__root__decreasing,axiom,
    ! [X: real,N_6: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_6))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_6),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X))) ) ) ) ).

tff(fact_3686_real__root__le__1__iff,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),one_one_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ) ).

tff(fact_3687_real__root__ge__1__iff,axiom,
    ! [Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) ) ) ).

tff(fact_3688_real__root__strict__decreasing,axiom,
    ! [X: real,N_6: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_6))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_6),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X))) ) ) ) ).

tff(fact_3689_real__root__lt__1__iff,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),one_one_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ) ).

tff(fact_3690_real__root__gt__1__iff,axiom,
    ! [Y_2: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) ) ) ).

tff(fact_3691_real__root__pos,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(suc,N_1))) = X ) ) ).

tff(fact_3692_real__root__less__mono__lemma,axiom,
    ! [Y: real,X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y))) ) ) ) ).

tff(fact_3693_real__root__pos__pos,axiom,
    ! [X_4: real,N_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_7))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_4))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_7),X_4))) ) ) ).

tff(fact_3694_real__root__mult__lemma,axiom,
    ! [Y: real,X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y)) ) ) ) ) ).

tff(fact_3695_real__root__pos__mult__exp,axiom,
    ! [X: real,N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M_1),hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) ) ) ) ) ).

tff(fact_3696_real__root__pos__unique,axiom,
    ! [X: real,Y: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N_1) = X )
         => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X) = Y ) ) ) ) ).

tff(fact_3697_real__root__pow__pos2,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),N_1) = X ) ) ) ).

tff(fact_3698_real__root__pos2,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1)) = X ) ) ) ).

tff(fact_3699_real__root__pow__pos,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),N_1) = X ) ) ) ).

tff(fact_3700_real__root__increasing,axiom,
    ! [X: real,N_6: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_6))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_6),X))) ) ) ) ) ).

tff(fact_3701_real__root__strict__increasing,axiom,
    ! [X: real,N_6: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_6))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_6),X))) ) ) ) ) ).

tff(fact_3702_sqrt__def,axiom,
    sqrt = hAPP_n546711566l_real(root,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3703_real__root__inverse__lemma,axiom,
    ! [X: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X)) ) ) ) ).

tff(fact_3704_DERIV__odd__real__root,axiom,
    ! [X_1: real,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ( ( X_1 != zero_zero_real )
       => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).

tff(fact_3705_negDivAlg_Opsimps,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba)))
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
         => ( negDivAlg(Aa,Ba) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)) ) )
        & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba)))
              | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
         => ( negDivAlg(Aa,Ba) = hAPP_P1975530577nt_int(adjust(Ba),negDivAlg(Aa,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Ba))) ) ) ) ) ).

tff(fact_3706_aux4,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))) ) ) ).

tff(fact_3707_MVT2,axiom,
    ! [F: fun_real_real,F_1: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => hBOOL(hAPP_real_bool(deriv_real(F,X_2),hAPP_real_real(F_1,X_2))) )
       => ? [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Z))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),Ba))
            & ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ba),Aa)),hAPP_real_real(F_1,Z)) ) ) ) ) ).

tff(fact_3708_polar__ex1,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ? [R_1: real,A_1: real] :
          ( ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_1)) )
          & ( Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_1)) ) ) ) ).

tff(fact_3709_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y))) ) ).

tff(fact_3710_exp__le__cancel__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ).

tff(fact_3711_exp__less__cancel__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ).

tff(fact_3712_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) ) ).

tff(fact_3713_exp__inj__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(exp_real,X_1) = hAPP_real_real(exp_real,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3714_abs__exp__cancel,axiom,
    ! [X: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X)) = hAPP_real_real(exp_real,X) ).

tff(fact_3715_ln__unique,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(exp_real,Y) = X )
     => ( hAPP_real_real(ln,X) = Y ) ) ).

tff(fact_3716_exp__ln__eq,axiom,
    ! [U_1: real,X: real] :
      ( ( hAPP_real_real(exp_real,U_1) = X )
     => ( hAPP_real_real(ln,X) = U_1 ) ) ).

tff(fact_3717_ln__exp,axiom,
    ! [X: real] : hAPP_real_real(ln,hAPP_real_real(exp_real,X)) = X ).

tff(fact_3718_not__exp__le__zero,axiom,
    ! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),zero_zero_real)) ).

tff(fact_3719_exp__ge__zero,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X))) ).

tff(fact_3720_exp__gt__zero,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X))) ).

tff(fact_3721_not__exp__less__zero,axiom,
    ! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),zero_zero_real)) ).

tff(fact_3722_exp__eq__one__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(exp_real,X_1) = one_one_real )
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_3723_one__le__exp__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(exp_real,X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) ) ).

tff(fact_3724_exp__le__one__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).

tff(fact_3725_exp__gt__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X))) ) ).

tff(fact_3726_exp__less__one__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ).

tff(fact_3727_one__less__exp__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_3728_exp__ge__add__one__self,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))) ).

tff(fact_3729_exp__real__of__nat__mult,axiom,
    ! [N_1: nat,X: real] : hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(exp_real,X)),N_1) ).

tff(fact_3730_exp__ln__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(exp_real,hAPP_real_real(ln,X_1)) = X_1 )
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_3731_exp__ln,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(exp_real,hAPP_real_real(ln,X)) = X ) ) ).

tff(fact_3732_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))) ) ).

tff(fact_3733_ln__x__over__x__mono,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,one_one_real)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,Y)),Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),X))) ) ) ).

tff(fact_3734_exp__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).

tff(fact_3735_negDivAlg_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [A_1: int,B_3: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3)))
           => ( ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_3)))
                    | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),zero_zero_int)) )
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_3))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_1),B_3)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_3736_posDivAlg_Opsimps,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba)))
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
         => ( posDivAlg(Aa,Ba) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),Aa) ) )
        & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
              | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
         => ( posDivAlg(Aa,Ba) = hAPP_P1975530577nt_int(adjust(Ba),posDivAlg(Aa,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Ba))) ) ) ) ) ).

tff(fact_3737_lemma__exp__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),one_one_real)))
          & ( hAPP_real_real(exp_real,X_2) = Y ) ) ) ).

tff(fact_3738_posDivAlg_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [A_1: int,B_3: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3)))
           => ( ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_3))
                    | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),zero_zero_int)) )
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_3))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_1),B_3)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_3739_log__base__10__eq1,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(ln,X)) ) ) ).

tff(fact_3740_log__base__10__eq2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,X)) ) ) ).

tff(fact_3741_exp__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X_2: real] : hAPP_real_real(exp_real,X_2) = Y ) ).

tff(fact_3742_polar__ex2,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real))
     => ? [R_1: real,A_1: real] :
          ( ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_1)) )
          & ( Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_1)) ) ) ) ).

tff(fact_3743_log__one,axiom,
    ! [A_2: real] : hAPP_real_real(log(A_2),one_one_real) = zero_zero_real ).

tff(fact_3744_Log_Olog__def,axiom,
    ! [A_2: real,X: real] : hAPP_real_real(log(A_2),X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,A_2)) ).

tff(fact_3745_log__less__cancel__iff,axiom,
    ! [Y_2: real,X_1: real,Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Aa))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(Aa),X_1)),hAPP_real_real(log(Aa),Y_2)))
          <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ) ) ).

tff(fact_3746_log__eq__one,axiom,
    ! [A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hAPP_real_real(log(A_2),A_2) = one_one_real ) ) ) ).

tff(fact_3747_log__ln,axiom,
    ! [X: real] : hAPP_real_real(ln,X) = hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X) ).

tff(fact_3748_log__le__cancel__iff,axiom,
    ! [Y_2: real,X_1: real,Aa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Aa))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(log(Aa),X_1)),hAPP_real_real(log(Aa),Y_2)))
          <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ) ) ).

tff(fact_3749_log__nat__power,axiom,
    ! [B_1: real,N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(log(B_1),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(log(B_1),X)) ) ) ).

tff(fact_3750_log__mult,axiom,
    ! [Y: real,X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
           => ( hAPP_real_real(log(A_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(log(A_2),X)),hAPP_real_real(log(A_2),Y)) ) ) ) ) ) ).

tff(fact_3751_log__divide,axiom,
    ! [Y: real,X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
           => ( hAPP_real_real(log(A_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_2),X)),hAPP_real_real(log(A_2),Y)) ) ) ) ) ) ).

tff(fact_3752_log__inverse,axiom,
    ! [X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hAPP_real_real(log(A_2),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_2),X)) ) ) ) ) ).

tff(fact_3753_log__eq__div__ln__mult__log,axiom,
    ! [X: real,B_1: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_1))
         => ( ( B_1 != one_one_real )
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
             => ( hAPP_real_real(log(A_2),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,B_1)),hAPP_real_real(ln,A_2))),hAPP_real_real(log(B_1),X)) ) ) ) ) ) ) ).

tff(fact_3754_two__times__odd__div__two__plus__one,axiom,
    ! [X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int) = X ) ) ).

tff(fact_3755_DERIV__isconst__all,axiom,
    ! [X_1: real,Y_2: real,F: fun_real_real] :
      ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real))
     => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_2) ) ) ).

tff(fact_3756_Code__Numeral_Oint__of__code,axiom,
    ! [K_2: code_code_numeral] :
      ( ( ( K_2 = zero_z126310315umeral )
       => ( code_int_of(K_2) = zero_zero_int ) )
      & ( ( K_2 != zero_z126310315umeral )
       => ( ( ( hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral )
           => ( code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) )
          & ( ( hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z126310315umeral )
           => ( code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) ) ) ) ) ) ).

tff(fact_3757_inv__inv,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),P_3))
           => ( hAPP_int_int(inv(P_3),hAPP_int_int(inv(P_3),A_2)) = A_2 ) ) ) ) ) ).

tff(fact_3758_even__neg,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,X_1)))
    <=> hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) ) ).

tff(fact_3759_code__numeral__zero__minus__one,axiom,
    hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,zero_z126310315umeral),one_on1645066479umeral) = zero_z126310315umeral ).

tff(fact_3760_even__difference,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) )
        | ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ) ).

tff(fact_3761_Parity_Oodd__plus__odd,axiom,
    ! [Y: int,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_3762_odd__plus__even,axiom,
    ! [Y: int,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_3763_Parity_Oeven__plus__odd,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_3764_Parity_Oeven__plus__even,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_3765_even__sum,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) )
        | ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ) ).

tff(fact_3766_Parity_Oodd__times__odd,axiom,
    ! [Y: int,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_3767_anything__times__even,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
     => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ).

tff(fact_3768_even__times__anything,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ).

tff(fact_3769_Parity_Oeven__product,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)))
    <=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
        | hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ).

tff(fact_3770_odd__one__int,axiom,
    ~ hBOOL(hAPP_int_bool(even_odd_even_int,one_one_int)) ).

tff(fact_3771_even__zero__int,axiom,
    hBOOL(hAPP_int_bool(even_odd_even_int,zero_zero_int)) ).

tff(fact_3772_odd__pow,axiom,
    ! [N_1: nat,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N_1))) ) ).

tff(fact_3773_even__power,axiom,
    ! [X_1: int,N: nat] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N)))
    <=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
        & ( N != zero_zero_nat ) ) ) ).

tff(fact_3774_zero__code__numeral__code,axiom,
    zero_z126310315umeral = hAPP_i769753017umeral(number1443263063umeral,pls) ).

tff(fact_3775_zprime__2,axiom,
    hBOOL(hAPP_int_bool(zprime,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3776_inv__less__p__minus__1,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(inv(P_3),A_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) ) ) ) ).

tff(fact_3777_inv__g__1,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(inv(P_3),A_2))) ) ) ) ).

tff(fact_3778_inv__not__p__minus__1,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( hAPP_int_int(inv(P_3),A_2) != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) ) ) ) ) ).

tff(fact_3779_inv__not__1,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( hAPP_int_int(inv(P_3),A_2) != one_one_int ) ) ) ) ).

tff(fact_3780_inv__distinct,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( A_2 != hAPP_int_int(inv(P_3),A_2) ) ) ) ) ).

tff(fact_3781_pos__int__even__equiv__nat__even,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
      <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat_1,X_1))) ) ) ).

tff(fact_3782_prime__g__5,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ( P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) )
       => ( ( P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) ) ) ) ).

tff(fact_3783_inv__not__0,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( hAPP_int_int(inv(P_3),A_2) != zero_zero_int ) ) ) ) ).

tff(fact_3784_even__equiv__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> ? [Y_1: int] : X_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1) ) ).

tff(fact_3785_zprime__zOdd__eq__grt__2,axiom,
    ! [Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Pa),zOdd))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa)) ) ) ).

tff(fact_3786_even__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ) ).

tff(fact_3787_two__times__even__div__two,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X ) ) ).

tff(fact_3788_one__code__numeral__code,axiom,
    one_on1645066479umeral = hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit1,pls)) ).

tff(fact_3789_odd__equiv__def,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> ? [Y_1: int] : X_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1)),one_one_int) ) ).

tff(fact_3790_odd__plus__one__div__two,axiom,
    ! [X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) ) ) ).

tff(fact_3791_even__plus__one__div__two,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3792_div__mod__code__numeral__def,axiom,
    ! [N_1: code_code_numeral,M_1: code_code_numeral] : code_d418564891umeral(N_1,M_1) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N_1),M_1)),hAPP_C498520661umeral(div_mo1740067990umeral(N_1),M_1)) ).

tff(fact_3793_wset__mem__inv__mem,axiom,
    ! [Ba: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Pa))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa)))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(Pa),Ba)),hAPP_i1948725293t_bool(wset(Aa),Pa))) ) ) ) ) ).

tff(fact_3794_wset__inv__mem__mem,axiom,
    ! [Ba: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Pa))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ba))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
             => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(Pa),Ba)),hAPP_i1948725293t_bool(wset(Aa),Pa)))
               => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa))) ) ) ) ) ) ) ).

tff(fact_3795_Euler__part2,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ) ).

tff(fact_3796_zcong__id,axiom,
    ! [M_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M_1),zero_zero_int),M_1)) ).

tff(fact_3797_IntPrimes_Ozcong__zero,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),zero_zero_int))
    <=> ( Aa = Ba ) ) ).

tff(fact_3798_zcong__zmod,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(div_mod_int(Aa),M)),hAPP_int_int(div_mod_int(Ba),M)),M)) ) ).

tff(fact_3799_zcong__1,axiom,
    ! [A_2: int,B_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),one_one_int)) ).

tff(fact_3800_zcong__zmult,axiom,
    ! [C_1: int,D_3: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_3)),M_1)) ) ) ).

tff(fact_3801_zcong__scalar2,axiom,
    ! [K_2: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),A_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),B_1)),M_1)) ) ).

tff(fact_3802_zcong__scalar,axiom,
    ! [K_2: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),K_2)),M_1)) ) ).

tff(fact_3803_zcong__zmult__prop2,axiom,
    ! [C: int,D: int,Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Aa)),M))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Ba)),M)) ) ) ).

tff(fact_3804_zcong__zmult__prop1,axiom,
    ! [C: int,D: int,Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),D)),M))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D)),M)) ) ) ).

tff(fact_3805_zcong__zmult__self,axiom,
    ! [A_2: int,M_1: int,B_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),M_1)),M_1)) ).

tff(fact_3806_zcong__zadd,axiom,
    ! [C_1: int,D_3: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),D_3)),M_1)) ) ) ).

tff(fact_3807_zcong__shift,axiom,
    ! [C_1: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),C_1)),M_1)) ) ).

tff(fact_3808_zcong__zdiff,axiom,
    ! [C_1: int,D_3: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),D_3)),M_1)) ) ) ).

tff(fact_3809_zcong__zminus,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),hAPP_int_int(uminus_uminus_int,M)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M)) ) ).

tff(fact_3810_zcong__zpower,axiom,
    ! [Z_1: nat,X: int,Y: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),Z_1)),M_1)) ) ).

tff(fact_3811_zcong__sym,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Ba),Aa),M)) ) ).

tff(fact_3812_zcong__refl,axiom,
    ! [K_2: int,M_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M_1)) ).

tff(fact_3813_zcong__eq__trans,axiom,
    ! [D_3: int,C_1: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => ( ( B_1 = C_1 )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),D_3),M_1)) ) ) ) ).

tff(fact_3814_zcong__trans,axiom,
    ! [C_1: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_1),C_1),M_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),C_1),M_1)) ) ) ).

tff(fact_3815_zcong__not,axiom,
    ! [B_1: int,M_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),M_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_2))
           => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1)) ) ) ) ) ).

tff(fact_3816_zcong__not__zero,axiom,
    ! [M_1: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M_1))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M_1)) ) ) ).

tff(fact_3817_zcong__less__eq,axiom,
    ! [M_1: int,Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M_1))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M_1))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M_1))
               => ( X = Y ) ) ) ) ) ) ) ).

tff(fact_3818_zcong__iff__lin,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
    <=> ? [K: int] : Ba = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),K)) ) ).

tff(fact_3819_Int2_Ozcong__zero,axiom,
    ! [M_1: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M_1))
         => ( X = zero_zero_int ) ) ) ) ).

tff(fact_3820_zcong__zless__0,axiom,
    ! [M_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),M_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),M_1))
         => ( A_2 = zero_zero_int ) ) ) ) ).

tff(fact_3821_zcong__zless__imp__eq,axiom,
    ! [B_1: int,M_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),M_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),M_1))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
             => ( A_2 = B_1 ) ) ) ) ) ) ).

tff(fact_3822_zcong__zmod__eq,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
      <=> ( hAPP_int_int(div_mod_int(Aa),M) = hAPP_int_int(div_mod_int(Ba),M) ) ) ) ).

tff(fact_3823_inv__not__p__minus__1__aux,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int))),one_one_int),Pa))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)),Pa)) ) ).

tff(fact_3824_zcong__zpower__zmult,axiom,
    ! [Z_1: nat,X: int,Y: nat,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),one_one_int),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1))),one_one_int),P_3)) ) ).

tff(fact_3825_wset__mem__mem,axiom,
    ! [Pa: int,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Aa))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),hAPP_i1948725293t_bool(wset(Aa),Pa))) ) ).

tff(fact_3826_zcong__zmult__prop3,axiom,
    ! [Y: int,X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3))
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),zero_zero_int),P_3)) ) ) ) ).

tff(fact_3827_zcong__zprime__prod__zero__contra,axiom,
    ! [B_1: int,A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
       => ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
            & ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_1),zero_zero_int),P_3)) )
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)),zero_zero_int),P_3)) ) ) ) ).

tff(fact_3828_zcong__zprime__prod__zero,axiom,
    ! [B_1: int,A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_1),zero_zero_int),P_3)) ) ) ) ) ).

tff(fact_3829_wset__subset,axiom,
    ! [Ba: int,Pa: int,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Aa))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),one_one_int)),Pa)))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa))) ) ) ).

tff(fact_3830_zcong__square,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),A_2)),one_one_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),one_one_int),P_3))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),P_3)) ) ) ) ) ).

tff(fact_3831_zcong__square__zless,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),A_2)),one_one_int),P_3))
           => ( ( A_2 = one_one_int )
              | ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) ) ) ) ) ) ) ).

tff(fact_3832_wset__g__1,axiom,
    ! [Ba: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ba)) ) ) ) ).

tff(fact_3833_wset__less,axiom,
    ! [Ba: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int))) ) ) ) ).

tff(fact_3834_inv__is__inv,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(inv(P_3),A_2))),one_one_int),P_3)) ) ) ) ).

tff(fact_3835_wset__mem__imp__or,axiom,
    ! [Ba: int,Pa: int,Aa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Aa))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),one_one_int)),Pa)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa)))
         => ( ( Ba = Aa )
            | ( Ba = hAPP_int_int(inv(Pa),Aa) ) ) ) ) ) ).

tff(fact_3836_Wilson__Russ,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_int(number_number_of_int,min)),P_3)) ) ).

tff(fact_3837_one__not__neg__one__mod__m,axiom,
    ! [M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(one_one_int),hAPP_int_int(number_number_of_int,min)),M_1)) ) ).

tff(fact_3838_zcong__neg__1__impl__ne__1,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(number_number_of_int,min)),P_3))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),one_one_int),P_3)) ) ) ).

tff(fact_3839_neg__one__power__eq__mod__m,axiom,
    ! [J_2: nat,K_2: nat,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)),M_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2) ) ) ) ).

tff(fact_3840_wset__mem,axiom,
    ! [Ba: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ba))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(wset(Aa),Pa))) ) ) ) ) ).

tff(fact_3841_d22set__eq__wset,axiom,
    ! [Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hAPP_i1948725293t_bool(d22set,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Pa) ) ) ).

tff(fact_3842_Euler__Criterion,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(legendre(A_2),P_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ).

tff(fact_3843_zfact__prop,axiom,
    ! [A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ) ) ).

tff(fact_3844_Euler__part1,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_int_int(number_number_of_int,min)),P_3)) ) ) ) ) ).

tff(fact_3845_Euler__part3,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int),P_3)) ) ) ) ) ).

tff(fact_3846_Legendre__def,axiom,
    ! [A_2: int,P_3: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
       => ( hAPP_int_int(legendre(A_2),P_3) = zero_zero_int ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_2))
           => ( hAPP_int_int(legendre(A_2),P_3) = one_one_int ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_2))
           => ( hAPP_int_int(legendre(A_2),P_3) = hAPP_int_int(number_number_of_int,min) ) ) ) ) ) ).

tff(fact_3847_QuadRes__def,axiom,
    ! [M: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,M),X_1))
    <=> ? [Y_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),M)) ) ).

tff(fact_3848_zcong__zless__unique,axiom,
    ! [A_2: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
     => ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),M_1))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),X_2),M_1))
          & ! [Y_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),Y_1),M_1)) )
             => ( Y_1 = X_2 ) ) ) ) ).

tff(fact_3849_nat__of__aux__code,axiom,
    ! [N_1: nat,I: code_code_numeral] :
      ( ( ( I = zero_z126310315umeral )
       => ( hAPP_nat_nat(code_nat_of_aux(I),N_1) = N_1 ) )
      & ( ( I != zero_z126310315umeral )
       => ( hAPP_nat_nat(code_nat_of_aux(I),N_1) = hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N_1)) ) ) ) ).

tff(fact_3850_mod__mod__is__mod,axiom,
    ! [X: int,M_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(div_mod_int(X),M_1)),M_1)) ).

tff(fact_3851_Residues_Oaux,axiom,
    ! [X: int,M_1: int,Y: int] :
      ( ( hAPP_int_int(div_mod_int(X),M_1) = hAPP_int_int(div_mod_int(Y),M_1) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M_1)) ) ).

tff(fact_3852_MultInvPair__distinct,axiom,
    ! [J_2: int,A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
           => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_2))
             => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ) ).

tff(fact_3853_MultInv__prop2a,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),X)),X)),one_one_int),P_3)) ) ) ) ).

tff(fact_3854_MultInv__prop2,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),one_one_int),P_3)) ) ) ) ).

tff(fact_3855_phi__prime,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( phi(P_3) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)) ) ) ).

tff(fact_3856_aux______1,axiom,
    ! [J_2: int,A_2: int,P_3: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),K_2))),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)) ) ).

tff(fact_3857_aux______3,axiom,
    ! [J_2: int,K_2: int,A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_2),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_2)),P_3)) ) ).

tff(fact_3858_MultInv__prop1,axiom,
    ! [X: int,Y: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3)) ) ) ).

tff(fact_3859_MultInv__zcong__prop1,axiom,
    ! [A_2: int,J_2: int,K_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),K_2))),P_3)) ) ) ).

tff(fact_3860_MultInv__def,axiom,
    ! [P_3: int,X: int] : hAPP_int_int(multInv(P_3),X) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3861_MultInv__prop5,axiom,
    ! [Y: int,X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3)) ) ) ) ) ) ).

tff(fact_3862_MultInv__prop4,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),X),P_3)) ) ) ) ).

tff(fact_3863_MultInv__prop3,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),zero_zero_int),P_3)) ) ) ) ).

tff(fact_3864_Int2_Oaux____1,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),P_3)) ) ) ) ).

tff(fact_3865_Int2_Oaux____2,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),X),P_3)) ) ) ) ).

tff(fact_3866_MultInv__zcong__prop3,axiom,
    ! [J_2: int,K_2: int,A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
           => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),K_2))),P_3))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3)) ) ) ) ) ) ) ).

tff(fact_3867_MultInv__zcong__prop2,axiom,
    ! [A_2: int,J_2: int,K_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),K_2))),P_3))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ) ).

tff(fact_3868_aux______2,axiom,
    ! [J_2: int,A_2: int,K_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_2),P_3)) ) ) ) ) ).

tff(fact_3869_aux______4,axiom,
    ! [K_2: int,A_2: int,J_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_2)),P_3))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ).

tff(fact_3870_SR__def,axiom,
    ! [Pa: int] : hAPP_i1948725293t_bool(sr,Pa) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Pa))) ).

tff(fact_3871_StandardRes__prop4,axiom,
    ! [X: int,Y: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(standardRes(M_1),X)),hAPP_int_int(standardRes(M_1),Y))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M_1)) ) ).

tff(fact_3872_complex__inverse,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(invers1449016382omplex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,B_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_3873_negative__zle,axiom,
    ! [N_1: nat,M_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))),hAPP_nat_int(semiri1621563631at_int,M_1))) ).

tff(fact_3874_negative__zless,axiom,
    ! [N_1: nat,M_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))),hAPP_nat_int(semiri1621563631at_int,M_1))) ).

tff(fact_3875_complex__zero__def,axiom,
    zero_zero_complex = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),zero_zero_real) ).

tff(fact_3876_Complex__eq__0,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,Aa),Ba) = zero_zero_complex )
    <=> ( ( Aa = zero_zero_real )
        & ( Ba = zero_zero_real ) ) ) ).

tff(fact_3877_complex__divide__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),hAPP_complex_complex(invers1449016382omplex,Y)) ).

tff(fact_3878_abs__int__eq,axiom,
    ! [M_1: nat] : hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M_1)) = hAPP_nat_int(semiri1621563631at_int,M_1) ).

tff(fact_3879_real__of__nat__def,axiom,
    real_nat = semiri132038758t_real ).

tff(fact_3880_real__eq__of__nat,axiom,
    real_nat = semiri132038758t_real ).

tff(fact_3881_nat__int,axiom,
    ! [N_1: nat] : hAPP_int_nat(nat_1,hAPP_nat_int(semiri1621563631at_int,N_1)) = N_1 ).

tff(fact_3882_complex__add,axiom,
    ! [A_2: real,B_1: real,C_1: real,D_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_2),C_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),D_3)) ).

tff(fact_3883_complex_Oinject,axiom,
    ! [Real1_2: real,Real2_2: real,Real1_1: real,Real2_1: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1_2),Real2_2) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1_1),Real2_1) )
    <=> ( ( Real1_2 = Real1_1 )
        & ( Real2_2 = Real2_1 ) ) ) ).

tff(fact_3884_complex__minus,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(uminus473333897omplex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,A_2)),hAPP_real_real(uminus_uminus_real,B_1)) ).

tff(fact_3885_not__neg__int,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N_1))) ).

tff(fact_3886_complex__diff,axiom,
    ! [A_2: real,B_1: real,C_1: real,D_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_2),C_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),D_3)) ).

tff(fact_3887_StandardRes__SR__prop,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sr,Pa)))
     => ( hAPP_int_int(standardRes(Pa),X_1) = X_1 ) ) ).

tff(fact_3888_StandardRes__prop1,axiom,
    ! [X: int,M_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(standardRes(M_1),X)),M_1)) ).

tff(fact_3889_StandardRes__def,axiom,
    ! [M_1: int,X: int] : hAPP_int_int(standardRes(M_1),X) = hAPP_int_int(div_mod_int(X),M_1) ).

tff(fact_3890_int__eq__0__conv,axiom,
    ! [N: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,N) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

tff(fact_3891_int__0,axiom,
    hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).

tff(fact_3892_transfer__int__nat__numerals_I1_J,axiom,
    zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).

tff(fact_3893_int__less__0__conv,axiom,
    ! [K_2: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_2)),zero_zero_int)) ).

tff(fact_3894_zero__zle__int,axiom,
    ! [N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) ).

tff(fact_3895_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
    ! [Z_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1))) ).

tff(fact_3896_transfer__int__nat__quantifiers_I2_J,axiom,
    ! [P_1: fun_int_bool] :
      ( ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(P_1,X_2)) )
    <=> ? [X_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_3897_transfer__int__nat__quantifiers_I1_J,axiom,
    ! [P_1: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_int_bool(P_1,X_2)) )
    <=> ! [X_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_3898_zless__int,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_3899_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ).

tff(fact_3900_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_3901_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ).

tff(fact_3902_zmult__int,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) ).

tff(fact_3903_int__mult,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) ).

tff(fact_3904_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ).

tff(fact_3905_zle__iff__zadd,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))
    <=> ? [N_2: nat] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_2)) ) ).

tff(fact_3906_zadd__int__left,axiom,
    ! [M_1: nat,N_1: nat,Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N_1)),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1))),Z_1) ).

tff(fact_3907_zadd__int,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) ).

tff(fact_3908_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ).

tff(fact_3909_int__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).

tff(fact_3910_transfer__int__nat__numerals_I2_J,axiom,
    one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).

tff(fact_3911_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

tff(fact_3912_not__int__zless__negative,axiom,
    ! [N_1: nat,M_1: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M_1)))) ).

tff(fact_3913_Complex__eq__number__of,axiom,
    ! [Aa: real,Ba: real,W_1: int] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,Aa),Ba) = hAPP_int_complex(number528085621omplex,W_1) )
    <=> ( ( Aa = hAPP_int_real(number267125858f_real,W_1) )
        & ( Ba = zero_zero_real ) ) ) ).

tff(fact_3914_Complex__eq__1,axiom,
    ! [Aa: real,Ba: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,Aa),Ba) = one_one_complex )
    <=> ( ( Aa = one_one_real )
        & ( Ba = zero_zero_real ) ) ) ).

tff(fact_3915_complex__one__def,axiom,
    one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex_1,one_one_real),zero_zero_real) ).

tff(fact_3916_real__of__int__of__nat__eq,axiom,
    ! [N_1: nat] : hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_real(real_nat,N_1) ).

tff(fact_3917_Divides_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y)) ).

tff(fact_3918_zdiv__int,axiom,
    ! [A_2: nat,B_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),B_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,A_2)),hAPP_nat_int(semiri1621563631at_int,B_1)) ).

tff(fact_3919_Divides_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X),Y)) ).

tff(fact_3920_zmod__int,axiom,
    ! [A_2: nat,B_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(A_2),B_1)) = hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,A_2)),hAPP_nat_int(semiri1621563631at_int,B_1)) ).

tff(fact_3921_zpower__int,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_1)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_1),N_1)) ).

tff(fact_3922_int__power,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_1),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_1)),N_1) ).

tff(fact_3923_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
    ! [X: nat,N_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1)) ).

tff(fact_3924_not__neg__nat,axiom,
    ! [Z_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,Z_1))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = Z_1 ) ) ).

tff(fact_3925_floor__real__of__nat,axiom,
    ! [N_1: nat] : archim1246769320r_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).

tff(fact_3926_Parity_Otransfer__int__nat__relations,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_1)))
    <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) ) ).

tff(fact_3927_even__nat__def,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).

tff(fact_3928_ceiling__real__of__nat,axiom,
    ! [N_1: nat] : archim856651990g_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).

tff(fact_3929_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).

tff(fact_3930_StandardRes__ubound,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(standardRes(P_3),X)),P_3)) ) ).

tff(fact_3931_StandardRes__prop3,axiom,
    ! [X_1: int,Pa: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),Pa))
    <=> ( hAPP_int_int(standardRes(Pa),X_1) != zero_zero_int ) ) ).

tff(fact_3932_StandardRes__eq__zcong,axiom,
    ! [M: int,X_1: int] :
      ( ( hAPP_int_int(standardRes(M),X_1) = zero_zero_int )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),M)) ) ).

tff(fact_3933_int__le__0__conv,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),zero_zero_int))
    <=> ( N = zero_zero_nat ) ) ).

tff(fact_3934_int__Suc0__eq__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).

tff(fact_3935_zless__iff__Suc__zadd,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
    <=> ? [N_2: nat] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_3936_int__Suc,axiom,
    ! [M_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_1)) ).

tff(fact_3937_nat__0__le,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = Z_1 ) ) ).

tff(fact_3938_int__eq__iff,axiom,
    ! [M: nat,Z_2: int] :
      ( ( hAPP_nat_int(semiri1621563631at_int,M) = Z_2 )
    <=> ( ( M = hAPP_int_nat(nat_1,Z_2) )
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) ) ) ).

tff(fact_3939_int__nat__eq,axiom,
    ! [Z_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = Z_1 ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = zero_zero_int ) ) ) ).

tff(fact_3940_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_int_nat(nat_1,Z_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M)),Z_2)) ) ).

tff(fact_3941_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

tff(fact_3942_negative__zle__0,axiom,
    ! [N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))),zero_zero_int)) ).

tff(fact_3943_nat__zminus__int,axiom,
    ! [N_1: nat] : hAPP_int_nat(nat_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1))) = zero_zero_nat ).

tff(fact_3944_zdiff__int,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) ) ) ).

tff(fact_3945_complex__mult,axiom,
    ! [A_2: real,B_1: real,C_1: real,D_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),C_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),D_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),D_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C_1))) ).

tff(fact_3946_neg__zminus__int,axiom,
    ! [N_1: nat] : hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))))) ).

tff(fact_3947_zero__less__int__conv,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_3948_zmult__zless__mono2__lemma,axiom,
    ! [K_2: nat,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),J_2))) ) ) ).

tff(fact_3949_transfer__int__nat__numerals_I4_J,axiom,
    hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3950_split__nat,axiom,
    ! [P_1: fun_nat_bool,I_1: int] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat_1,I_1)))
    <=> ( ! [N_2: nat] :
            ( ( I_1 = hAPP_nat_int(semiri1621563631at_int,N_2) )
           => hBOOL(hAPP_nat_bool(P_1,N_2)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int))
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ).

tff(fact_3951_nat__eq__iff,axiom,
    ! [W_1: int,M: nat] :
      ( ( hAPP_int_nat(nat_1,W_1) = M )
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( W_1 = hAPP_nat_int(semiri1621563631at_int,M) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( M = zero_zero_nat ) ) ) ) ).

tff(fact_3952_nat__eq__iff2,axiom,
    ! [M: nat,W_1: int] :
      ( ( M = hAPP_int_nat(nat_1,W_1) )
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( W_1 = hAPP_nat_int(semiri1621563631at_int,M) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( M = zero_zero_nat ) ) ) ) ).

tff(fact_3953_negative__zless__0,axiom,
    ! [N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))),zero_zero_int)) ).

tff(fact_3954_not__zle__0__negative,axiom,
    ! [N_1: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))))) ).

tff(fact_3955_int__eq__iff__number__of,axiom,
    ! [M: nat,V: int] :
      ( ( hAPP_nat_int(semiri1621563631at_int,M) = hAPP_int_int(number_number_of_int,V) )
    <=> ( ( M = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,V)) )
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V))) ) ) ).

tff(fact_3956_int__nat__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_int ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_int(number_number_of_int,V_2) ) ) ) ).

tff(fact_3957_StandardRes__lbound,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(standardRes(P_3),X))) ) ).

tff(fact_3958_StandardRes__prop2,axiom,
    ! [X1_1: int,X2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(standardRes(M),X1_1) = hAPP_int_int(standardRes(M),X2) )
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X1_1),X2),M)) ) ) ).

tff(fact_3959_floor__minus__real__of__nat,axiom,
    ! [N_1: nat] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N_1))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1)) ).

tff(fact_3960_transfer__int__nat__numerals_I3_J,axiom,
    hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3961_nat__less__iff,axiom,
    ! [M: nat,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),M))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_nat_int(semiri1621563631at_int,M))) ) ) ).

tff(fact_3962_zdiff__int__split,axiom,
    ! [P_1: fun_int_bool,X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_2))))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1))
         => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))) )
        & ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
         => hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).

tff(fact_3963_SRStar__mult__prop2,axiom,
    ! [X_1: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(Pa),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),hAPP_int_int(multInv(Pa),X_1)))),hAPP_i1948725293t_bool(sRStar,Pa))) ) ) ) ) ).

tff(fact_3964_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3965_int__if__cong,axiom,
    ! [X_1: nat,Y_2: nat,P_1: bool] :
      ( ( hBOOL(P_1)
       => ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_1),Y_2)) ) )
      & ( ~ hBOOL(P_1)
       => ( hAPP_nat_int(semiri1621563631at_int,Y_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_1),Y_2)) ) ) ) ).

tff(fact_3966_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,M) = hAPP_nat_int(semiri1621563631at_int,N) )
    <=> ( M = N ) ) ).

tff(fact_3967_complex__diff__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),hAPP_complex_complex(uminus473333897omplex,Y)) ).

tff(fact_3968_real__norm__def,axiom,
    ! [R_3: real] : hAPP_real_real(norm_norm_real,R_3) = hAPP_real_real(abs_abs_real,R_3) ).

tff(fact_3969_StandardRes__SRStar__prop3,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
     => ( hAPP_int_int(standardRes(Pa),X_1) = X_1 ) ) ).

tff(fact_3970_complex__mod__triangle__ineq2,axiom,
    ! [B_1: complex,A_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_2))),hAPP_complex_real(norm_norm_complex,B_1))),hAPP_complex_real(norm_norm_complex,A_2))) ).

tff(fact_3971_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X))),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_3972_SRStar__SR__prop,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sr,Pa))) ) ).

tff(fact_3973_SRStar__def,axiom,
    ! [Pa: int] : hAPP_i1948725293t_bool(sRStar,Pa) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Pa))) ).

tff(fact_3974_StandardRes__SRStar__prop1a,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),Pa)) ) ).

tff(fact_3975_cmod__unit__one,axiom,
    ! [A_2: real] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2))) = one_one_real ).

tff(fact_3976_complex__norm,axiom,
    ! [X: real,Y: real] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3977_StandardRes__SRStar__prop4,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(Pa),X_1)),hAPP_i1948725293t_bool(sRStar,Pa))) ) ) ) ).

tff(fact_3978_StandardRes__SRStar__prop1,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(Pa),X_1)),hAPP_i1948725293t_bool(sRStar,Pa)))
      <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),Pa)) ) ) ).

tff(fact_3979_SRStar__mult__prop1,axiom,
    ! [Y_2: int,X_1: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),hAPP_i1948725293t_bool(sRStar,Pa)))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(Pa),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2))),hAPP_i1948725293t_bool(sRStar,Pa))) ) ) ) ) ).

tff(fact_3980_StandardRes__SRStar__prop2,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => ( hBOOL(hAPP_int_bool(zprime,Pa))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,Pa)))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(Pa),hAPP_int_int(multInv(Pa),X_1))),hAPP_i1948725293t_bool(sRStar,Pa))) ) ) ) ).

tff(fact_3981_negD,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),zero_zero_int))
     => ? [N_2: nat] : X = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_3982_zero__less__imp__eq__int,axiom,
    ! [K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
     => ? [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
          & ( K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) ) ) ) ).

tff(fact_3983_complex_Osize_I1_J,axiom,
    ! [Real1: real,Real2: real] : complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1),Real2)) = zero_zero_nat ).

tff(fact_3984_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) ) ) ) ).

tff(fact_3985_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

tff(fact_3986_bezw__non__0,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) ) ) ).

tff(fact_3987_zero__le__imp__eq__int,axiom,
    ! [K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ? [N_2: nat] : K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) ) ).

tff(fact_3988_nonneg__eq__int,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ~ ! [M_2: nat] : Z_1 != hAPP_nat_int(semiri1621563631at_int,M_2) ) ).

tff(fact_3989_nonneg__int__cases,axiom,
    ! [K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ~ ! [N_2: nat] : K_2 != hAPP_nat_int(semiri1621563631at_int,N_2) ) ).

tff(fact_3990_cos__arg__i__mult__zero__neg,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real))
     => ( hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),Y))) = zero_zero_real ) ) ).

tff(fact_3991_cos__arg__i__mult__zero__pos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ( hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),Y))) = zero_zero_real ) ) ).

tff(fact_3992_int__nat__abs,axiom,
    ! [X: int] : hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_3993_cos__arg__i__mult__zero,axiom,
    ! [Y: real] :
      ( ( Y != zero_zero_real )
     => ( hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),Y))) = zero_zero_real ) ) ).

tff(fact_3994_complex_Osize_I2_J,axiom,
    ! [Real1: real,Real2: real] : hAPP_complex_nat(size_size_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1),Real2)) = zero_zero_nat ).

tff(fact_3995_SRStar__card,axiom,
    ! [Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(sRStar,Pa))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int) ) ) ).

tff(fact_3996_int__diff__cases,axiom,
    ! [Z_1: int] :
      ~ ! [M_2: nat,N_2: nat] : Z_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_2)) ).

tff(fact_3997_MultInvPair__card__two,axiom,
    ! [J: int,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Pa),Aa))
           => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J),zero_zero_int),Pa))
             => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(multInvPair(Aa,Pa),J)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ) ).

tff(fact_3998_SetS__elems__card,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Pa),Aa))
           => ! [X_2: fun_int_bool] :
                ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(Aa,Pa)))
               => ( hAPP_f957591787ol_nat(finite_card_int,X_2) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ) ).

tff(fact_3999_power2__i,axiom,
    hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,ii),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_complex(number528085621omplex,min) ).

tff(fact_4000_complex__i__not__zero,axiom,
    ii != zero_zero_complex ).

tff(fact_4001_complex__i__not__one,axiom,
    ii != one_one_complex ).

tff(fact_4002_complex__i__not__number__of,axiom,
    ! [W: int] : ii != hAPP_int_complex(number528085621omplex,W) ).

tff(fact_4003_complex__i__mult__minus,axiom,
    ! [X: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X)) = hAPP_complex_complex(uminus473333897omplex,X) ).

tff(fact_4004_inverse__i,axiom,
    hAPP_complex_complex(invers1449016382omplex,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).

tff(fact_4005_i__def,axiom,
    ii = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),one_one_real) ).

tff(fact_4006_Complex__eq__i,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,X_1),Y_2) = ii )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = one_one_real ) ) ) ).

tff(fact_4007_Complex__mult__i,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,B_1)),A_2) ).

tff(fact_4008_i__mult__Complex,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,B_1)),A_2) ).

tff(fact_4009_i__squared,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_int_complex(number528085621omplex,min) ).

tff(fact_4010_i__mult__eq2,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).

tff(fact_4011_SetS__card,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Pa),Aa))
           => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(Aa,Pa))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ) ) ) ).

tff(fact_4012_complex__inverse__complex__split,axiom,
    ! [X: real,Y: real] : hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).

tff(fact_4013_MultInvPair__prop2,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( comple219730294t_bool(setS(Aa,Pa)) = hAPP_i1948725293t_bool(sRStar,Pa) ) ) ) ) ).

tff(fact_4014_complex__eq__cancel__iff2,axiom,
    ! [X_1: real,Y_2: real,Xa_1: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,X_1),Y_2) = of_real_complex(Xa_1) )
    <=> ( ( X_1 = Xa_1 )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_4015_complex__of__real__def,axiom,
    ! [R_3: real] : of_real_complex(R_3) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,R_3),zero_zero_real) ).

tff(fact_4016_sgn__eq,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(sgn_sgn_complex,Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Z_1),of_real_complex(hAPP_complex_real(norm_norm_complex,Z_1))) ).

tff(fact_4017_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),R_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),R_3)) ).

tff(fact_4018_complex__of__real__mult__Complex,axiom,
    ! [R_3: real,X: real,Y: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),Y)) ).

tff(fact_4019_complex__of__real__add__Complex,axiom,
    ! [R_3: real,X: real,Y: real] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),X)),Y) ).

tff(fact_4020_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),R_3)),Y) ).

tff(fact_4021_i__mult__eq,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = of_real_complex(hAPP_int_real(number267125858f_real,min)) ).

tff(fact_4022_i__complex__of__real,axiom,
    ! [R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),R_3) ).

tff(fact_4023_complex__of__real__i,axiom,
    ! [R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),R_3) ).

tff(fact_4024_complex__of__real__minus__one,axiom,
    of_real_complex(hAPP_real_real(uminus_uminus_real,one_one_real)) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).

tff(fact_4025_cmod__complex__polar,axiom,
    ! [R_3: real,A_2: real] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2)))) = hAPP_real_real(abs_abs_real,R_3) ).

tff(fact_4026_expi__two__pi__i,axiom,
    expi(hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),of_real_complex(pi))),ii)) = one_one_complex ).

tff(fact_4027_expi__add,axiom,
    ! [A_2: complex,B_1: complex] : expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_2),B_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_2)),expi(B_1)) ).

tff(fact_4028_expi__zero,axiom,
    expi(zero_zero_complex) = one_one_complex ).

tff(fact_4029_complex__split__polar,axiom,
    ! [Z_1: complex] :
    ? [R_1: real,A_1: real] : Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_1)),hAPP_real_real(sin,A_1))) ).

tff(fact_4030_Union__SetS__finite,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,comple219730294t_bool(setS(Aa,Pa)))) ) ).

tff(fact_4031_wset__fin,axiom,
    ! [Aa: int,Pa: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(Aa),Pa))) ).

tff(fact_4032_d22set__fin,axiom,
    ! [Aa: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,Aa))) ).

tff(fact_4033_SetS__elems__finite,axiom,
    ! [Aa: int,Pa: int,X_2: fun_int_bool] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(Aa,Pa)))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)) ) ).

tff(fact_4034_SRStar__finite,axiom,
    ! [Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(sRStar,Pa))) ) ).

tff(fact_4035_SetS__finite,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,setS(Aa,Pa))) ) ).

tff(fact_4036_DERIV__neg__imp__decreasing,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa))) ) ) ).

tff(fact_4037_DERIV__pos__imp__increasing,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Aa)),hAPP_real_real(F,Ba))) ) ) ).

tff(fact_4038_coprime__sos,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_4039_coprime__commute,axiom,
    ! [Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Aa),Ba))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Ba),Aa)) ) ).

tff(fact_4040_coprime__lmul2,axiom,
    ! [D_3: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_1)) ) ).

tff(fact_4041_coprime__rmul2,axiom,
    ! [D_3: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_2)) ) ).

tff(fact_4042_coprime__mul,axiom,
    ! [B_1: nat,D_3: nat,A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1))) ) ) ).

tff(fact_4043_coprime__mul__eq,axiom,
    ! [D: nat,Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),Ba)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),Aa))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),Ba)) ) ) ).

tff(fact_4044_coprime__1_H,axiom,
    ! [A_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_2)) ).

tff(fact_4045_coprime__1,axiom,
    ! [A_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),one_one_nat)) ).

tff(fact_4046_coprime__refl,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,N),N))
    <=> ( N = one_one_nat ) ) ).

tff(fact_4047_coprime__exp__imp,axiom,
    ! [N_1: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),B_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),N_1))) ) ).

tff(fact_4048_coprime__exp,axiom,
    ! [N_1: nat,D_3: nat,A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1))) ) ).

tff(fact_4049_coprime__Suc0_H,axiom,
    ! [A_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(suc,zero_zero_nat)),A_2)) ).

tff(fact_4050_coprime__Suc0,axiom,
    ! [A_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_4051_coprime__0,axiom,
    ! [D: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),zero_zero_nat))
    <=> ( D = one_one_nat ) ) ).

tff(fact_4052_coprime__0_H,axiom,
    ! [D: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D))
    <=> ( D = one_one_nat ) ) ).

tff(fact_4053_coprime__plus1,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),N_1)) ).

tff(fact_4054_coprime__exp2,axiom,
    ! [Aa: nat,N: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Aa),hAPP_nat_nat(suc,N))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ba),hAPP_nat_nat(suc,N))))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Aa),Ba)) ) ).

tff(fact_4055_coprime__minus1,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),N_1)) ) ).

tff(fact_4056_coprime__bezout,axiom,
    ! [Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Aa),Ba))
    <=> ? [X_2: nat,Y_1: nat] :
          ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),Y_1)) = one_one_nat )
          | ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),Y_1)) = one_one_nat ) ) ) ).

tff(fact_4057_nat_Osize_I2_J,axiom,
    ! [Nat: nat] : hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4058_coprime__bezout__strong,axiom,
    ! [A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),B_1))
     => ( ( B_1 != one_one_nat )
       => ? [X_2: nat,Y_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),Y_1)),one_one_nat) ) ) ).

tff(fact_4059_nat_Osize_I1_J,axiom,
    hAPP_nat_nat(nat_size,zero_zero_nat) = zero_zero_nat ).

tff(fact_4060_nat_Osize_I4_J,axiom,
    ! [Nat: nat] : hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4061_complex__expi__Ex,axiom,
    ! [Z_1: complex] :
    ? [A_1: complex,R_1: real] : Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_1)) ).

tff(fact_4062_ResSet__finite,axiom,
    ! [X_3: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_f448129468l_bool(resSet(M),X_3))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_3)) ) ) ).

tff(fact_4063_nat__size,axiom,
    ! [N_1: nat] : hAPP_nat_nat(size_size_nat,N_1) = N_1 ).

tff(fact_4064_ResSet__def,axiom,
    ! [M: int,X_3: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(resSet(M),X_3))
    <=> ! [Y1: int,Y2: int] :
          ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_3))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_3))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y1),Y2),M)) )
         => ( Y1 = Y2 ) ) ) ).

tff(fact_4065_nat_Osize_I3_J,axiom,
    hAPP_nat_nat(size_size_nat,zero_zero_nat) = zero_zero_nat ).

tff(fact_4066_ResSet__SRStar__prop,axiom,
    ! [Pa: int] : hBOOL(hAPP_f448129468l_bool(resSet(Pa),hAPP_i1948725293t_bool(sRStar,Pa))) ).

tff(fact_4067_code__numeral_Osize_I1_J,axiom,
    code_c271388182l_size(zero_z126310315umeral) = zero_zero_nat ).

tff(fact_4068_complex__mod__mult__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(norm_norm_complex,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_4069_complex__cnj__complex__of__real,axiom,
    ! [X: real] : hAPP_complex_complex(cnj,of_real_complex(X)) = of_real_complex(X) ).

tff(fact_4070_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4071_complex__cnj__power,axiom,
    ! [X: complex,N_1: nat] : hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X),N_1)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X)),N_1) ).

tff(fact_4072_complex__cnj__of__nat,axiom,
    ! [N_1: nat] : hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N_1)) = hAPP_nat_complex(semiri2020571505omplex,N_1) ).

tff(fact_4073_complex__cnj__one,axiom,
    hAPP_complex_complex(cnj,one_one_complex) = one_one_complex ).

tff(fact_4074_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4075_cnj_Oadd,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4076_complex__cnj__number__of,axiom,
    ! [W: int] : hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W)) = hAPP_int_complex(number528085621omplex,W) ).

tff(fact_4077_cnj_Ozero,axiom,
    hAPP_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

tff(fact_4078_complex__cnj__zero__iff,axiom,
    ! [Z_2: complex] :
      ( ( hAPP_complex_complex(cnj,Z_2) = zero_zero_complex )
    <=> ( Z_2 = zero_zero_complex ) ) ).

tff(fact_4079_complex__cnj__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1)) = Z_1 ).

tff(fact_4080_complex__cnj__cancel__iff,axiom,
    ! [X_1: complex,Y_2: complex] :
      ( ( hAPP_complex_complex(cnj,X_1) = hAPP_complex_complex(cnj,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_4081_complex__cnj__inverse,axiom,
    ! [X: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X)) ).

tff(fact_4082_complex__mod__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_real(norm_norm_complex,Z_1) ).

tff(fact_4083_cnj_Odiff,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4084_cnj_Ominus,axiom,
    ! [X: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X)) ).

tff(fact_4085_complex__cnj,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(cnj,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),hAPP_real_real(uminus_uminus_real,B_1)) ).

tff(fact_4086_complex__cnj__i,axiom,
    hAPP_complex_complex(cnj,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).

tff(fact_4087_cnj_Opos__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4088_cnj_Ononneg__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4089_cnj_Obounded,axiom,
    ? [K_4: real] :
    ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ).

tff(fact_4090_complex__diff__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(im,Z_1)))),ii) ).

tff(fact_4091_complex__add__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(re,Z_1))) ).

tff(fact_4092_code__numeral_Osize_I2_J,axiom,
    ! [Code_numeral_1: code_code_numeral] : code_c271388182l_size(code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,code_c271388182l_size(Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4093_Re__complex__of__real,axiom,
    ! [Z_1: real] : hAPP_complex_real(re,of_real_complex(Z_1)) = Z_1 ).

tff(fact_4094_complex__add__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) ).

tff(fact_4095_complex__Im__mult,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y))) ).

tff(fact_4096_code__numeral_Oinject,axiom,
    ! [Code_numeral_5: code_code_numeral,Code_numeral_4: code_code_numeral] :
      ( ( code_S1047413653umeral(Code_numeral_5) = code_S1047413653umeral(Code_numeral_4) )
    <=> ( Code_numeral_5 = Code_numeral_4 ) ) ).

tff(fact_4097_complex__eq__iff,axiom,
    ! [X_1: complex,Y_2: complex] :
      ( ( X_1 = Y_2 )
    <=> ( ( hAPP_complex_real(re,X_1) = hAPP_complex_real(re,Y_2) )
        & ( hAPP_complex_real(im,X_1) = hAPP_complex_real(im,Y_2) ) ) ) ).

tff(fact_4098_complex__eqI,axiom,
    ! [X: complex,Y: complex] :
      ( ( hAPP_complex_real(re,X) = hAPP_complex_real(re,Y) )
     => ( ( hAPP_complex_real(im,X) = hAPP_complex_real(im,Y) )
       => ( X = Y ) ) ) ).

tff(fact_4099_complex__surj,axiom,
    ! [Z_1: complex] : hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_complex_real(re,Z_1)),hAPP_complex_real(im,Z_1)) = Z_1 ).

tff(fact_4100_Re,axiom,
    ! [X: real,Y: real] : hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = X ).

tff(fact_4101_Im,axiom,
    ! [X: real,Y: real] : hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = Y ).

tff(fact_4102_complex__Re__mult,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) ).

tff(fact_4103_complex__minus__def,axiom,
    ! [X: complex] : hAPP_complex_complex(uminus473333897omplex,X) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X))),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))) ).

tff(fact_4104_cnj__def,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(cnj,Z_1) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_complex_real(re,Z_1)),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,Z_1))) ).

tff(fact_4105_complex__Re__cnj,axiom,
    ! [X: complex] : hAPP_complex_real(re,hAPP_complex_complex(cnj,X)) = hAPP_complex_real(re,X) ).

tff(fact_4106_complex__Re__le__cmod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(re,X)),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_4107_complex__Im__cnj,axiom,
    ! [X: complex] : hAPP_complex_real(im,hAPP_complex_complex(cnj,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) ).

tff(fact_4108_complex__Re__i,axiom,
    hAPP_complex_real(re,ii) = zero_zero_real ).

tff(fact_4109_Im__complex__of__real,axiom,
    ! [Z_1: real] : hAPP_complex_real(im,of_real_complex(Z_1)) = zero_zero_real ).

tff(fact_4110_complex__Im__i,axiom,
    hAPP_complex_real(im,ii) = one_one_real ).

tff(fact_4111_code__numeral_Osimps_I3_J,axiom,
    ! [Code_numeral_3: code_code_numeral] : code_S1047413653umeral(Code_numeral_3) != zero_z126310315umeral ).

tff(fact_4112_code__numeral_Osimps_I2_J,axiom,
    ! [Code_numeral_2: code_code_numeral] : zero_z126310315umeral != code_S1047413653umeral(Code_numeral_2) ).

tff(fact_4113_Re_Ozero,axiom,
    hAPP_complex_real(re,zero_zero_complex) = zero_zero_real ).

tff(fact_4114_Im_Ozero,axiom,
    hAPP_complex_real(im,zero_zero_complex) = zero_zero_real ).

tff(fact_4115_complex__Im__one,axiom,
    hAPP_complex_real(im,one_one_complex) = zero_zero_real ).

tff(fact_4116_Re_Oadd,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) ).

tff(fact_4117_Im_Oadd,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) ).

tff(fact_4118_complex__Re__one,axiom,
    hAPP_complex_real(re,one_one_complex) = one_one_real ).

tff(fact_4119_complex__Im__number__of,axiom,
    ! [V_2: int] : hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_2)) = zero_zero_real ).

tff(fact_4120_Re_Ominus,axiom,
    ! [X: complex] : hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X)) ).

tff(fact_4121_complex__Re__number__of,axiom,
    ! [V_2: int] : hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_2)) = hAPP_int_real(number267125858f_real,V_2) ).

tff(fact_4122_Im_Ominus,axiom,
    ! [X: complex] : hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) ).

tff(fact_4123_Re_Odiff,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) ).

tff(fact_4124_Im_Odiff,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) ).

tff(fact_4125_complex__mult__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y)))) ).

tff(fact_4126_complex__Im__of__nat,axiom,
    ! [N_1: nat] : hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N_1)) = zero_zero_real ).

tff(fact_4127_complex__Re__of__nat,axiom,
    ! [N_1: nat] : hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N_1)) = hAPP_nat_real(semiri132038758t_real,N_1) ).

tff(fact_4128_abs__Re__le__cmod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(re,X))),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_4129_abs__Im__le__cmod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X))),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_4130_complex__In__mult__cnj__zero,axiom,
    ! [Z_1: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = zero_zero_real ).

tff(fact_4131_Re__sgn,axiom,
    ! [Z_1: complex] : hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) ).

tff(fact_4132_Im__sgn,axiom,
    ! [Z_1: complex] : hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(im,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) ).

tff(fact_4133_cmod__def,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4134_complex__Re__inverse,axiom,
    ! [X: complex] : hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4135_Suc__code__numeral__minus__one,axiom,
    ! [N_1: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N_1)),one_on1645066479umeral) = N_1 ).

tff(fact_4136_complex__Im__inverse,axiom,
    ! [X: complex] : hAPP_complex_real(im,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4137_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) ).

tff(fact_4138_complex__mult__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4139_complex__inverse__def,axiom,
    ! [X: complex] : hAPP_complex_complex(invers1449016382omplex,X) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_4140_Im_Opos__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4141_Re_Opos__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4142_Im_Ononneg__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4143_Re_Ononneg__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4144_Im_Obounded,axiom,
    ? [K_4: real] :
    ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ).

tff(fact_4145_Re_Obounded,axiom,
    ? [K_4: real] :
    ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ).

tff(fact_4146_code__numeral_Osize_I4_J,axiom,
    ! [Code_numeral_1: code_code_numeral] : hAPP_C2129356693al_nat(size_s945831648umeral,code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_C2129356693al_nat(size_s945831648umeral,Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4147_code__numeral_Oexhaust,axiom,
    ! [Y: code_code_numeral] :
      ( ( Y != zero_z126310315umeral )
     => ~ ! [Code_numeral: code_code_numeral] : Y != code_S1047413653umeral(Code_numeral) ) ).

tff(fact_4148_expi__def,axiom,
    ! [Z_1: complex] : expi(Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(exp_real,hAPP_complex_real(re,Z_1)))),cis(hAPP_complex_real(im,Z_1))) ).

tff(fact_4149_Re__cis,axiom,
    ! [A_2: real] : hAPP_complex_real(re,cis(A_2)) = hAPP_real_real(cos,A_2) ).

tff(fact_4150_Im__cis,axiom,
    ! [A_2: real] : hAPP_complex_real(im,cis(A_2)) = hAPP_real_real(sin,A_2) ).

tff(fact_4151_cis__mult,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_2)),cis(B_1)) = cis(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_2),B_1)) ).

tff(fact_4152_cis__zero,axiom,
    cis(zero_zero_real) = one_one_complex ).

tff(fact_4153_cis__inverse,axiom,
    ! [A_2: real] : hAPP_complex_complex(invers1449016382omplex,cis(A_2)) = cis(hAPP_real_real(uminus_uminus_real,A_2)) ).

tff(fact_4154_cis__divide,axiom,
    ! [A_2: real,B_1: real] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,cis(A_2)),cis(B_1)) = cis(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_2),B_1)) ).

tff(fact_4155_cis__def,axiom,
    ! [A_2: real] : cis(A_2) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2)) ).

tff(fact_4156_DeMoivre,axiom,
    ! [A_2: real,N_1: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_2)),N_1) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_2)) ).

tff(fact_4157_cis__real__of__nat__Suc__mult,axiom,
    ! [N_1: nat,A_2: real] : cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))),A_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_2)),cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_2))) ).

tff(fact_4158_code__numeral_Osize_I3_J,axiom,
    hAPP_C2129356693al_nat(size_s945831648umeral,zero_z126310315umeral) = zero_zero_nat ).

tff(fact_4159_cos__n__Re__cis__pow__n,axiom,
    ! [N_1: nat,A_2: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_2)) = hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_2)),N_1)) ).

tff(fact_4160_sin__n__Im__cis__pow__n,axiom,
    ! [N_1: nat,A_2: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_2)) = hAPP_complex_real(im,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_2)),N_1)) ).

tff(fact_4161_divides__cases,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M_1))
     => ( ( M_1 = zero_zero_nat )
        | ( M_1 = N_1 )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),M_1)) ) ) ).

tff(fact_4162_dvd_Oorder__refl,axiom,
    ! [X: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ).

tff(fact_4163_dvd__1__left,axiom,
    ! [K_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2)) ).

tff(fact_4164_coprime__divisors,axiom,
    ! [E_1: nat,B_1: nat,D_3: nat,A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E_1),B_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),B_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),E_1)) ) ) ) ).

tff(fact_4165_divides__exp,axiom,
    ! [N_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),N_1))) ) ).

tff(fact_4166_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
    ! [A_2: nat] :
      ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),zero_zero_nat)) ) ).

tff(fact_4167_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
    ! [A_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),zero_zero_nat)) ).

tff(fact_4168_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Aa),zero_zero_nat))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),Aa)) ) ) ).

tff(fact_4169_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
    ! [Aa: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),Aa))
    <=> ( Aa = zero_zero_nat ) ) ).

tff(fact_4170_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
    ! [A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_2))
     => ( A_2 = zero_zero_nat ) ) ).

tff(fact_4171_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
    ! [A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),one_one_nat))
     => ( A_2 = one_one_nat ) ) ).

tff(fact_4172_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
    ! [Aa: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Aa),one_one_nat))
    <=> ( Aa = one_one_nat ) ) ).

tff(fact_4173_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
    ! [Aa: nat] :
      ( ( Aa != one_one_nat )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),Aa))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Aa),one_one_nat)) ) ) ).

tff(fact_4174_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
    ! [A_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_2)) ).

tff(fact_4175_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
    ! [A_2: nat] :
      ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),one_one_nat))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_2)) ) ).

tff(fact_4176_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

tff(fact_4177_divides__mul__r,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C_1))) ) ).

tff(fact_4178_divides__mul__l,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),A_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),B_1))) ) ).

tff(fact_4179_divides__add__revr,axiom,
    ! [B_1: nat,D_3: nat,A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),B_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_1)) ) ) ).

tff(fact_4180_dvd__reduce,axiom,
    ! [K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) ) ).

tff(fact_4181_dvd_Oless__irrefl,axiom,
    ! [X: nat] :
      ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ) ).

tff(fact_4182_dvd_Oeq__iff,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) ) ) ).

tff(fact_4183_dvd_Ole__less,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_4184_dvd_Oless__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_4185_dvd_Oless__le__not__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) ) ) ).

tff(fact_4186_dvd_Oneq__le__trans,axiom,
    ! [A_2: nat,B_1: nat] :
      ( ( A_2 != B_1 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A_2)) ) ) ) ).

tff(fact_4187_dvd_Oeq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).

tff(fact_4188_dvd_Oantisym__conv,axiom,
    ! [Y_2: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_4189_dvd_Ole__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
        | ( X = Y ) ) ) ).

tff(fact_4190_dvd_Ole__neq__trans,axiom,
    ! [A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
     => ( ( A_2 != B_1 )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A_2)) ) ) ) ).

tff(fact_4191_dvd_Oord__eq__le__trans,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( ( A_2 = B_1 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),C_1)) ) ) ).

tff(fact_4192_dvd_Oord__le__eq__trans,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
     => ( ( B_1 = C_1 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),C_1)) ) ) ).

tff(fact_4193_dvd__antisym,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M_1))
       => ( M_1 = N_1 ) ) ) ).

tff(fact_4194_dvd_Oantisym,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
       => ( X = Y ) ) ) ).

tff(fact_4195_dvd_Oorder__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) ) ) ).

tff(fact_4196_dvd_Oord__eq__less__trans,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( ( A_2 = B_1 )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),B_1)) )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),C_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_2)) ) ) ) ).

tff(fact_4197_dvd_Ole__less__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).

tff(fact_4198_dvd_Oless__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( X != Y ) ) ).

tff(fact_4199_dvd_Oless__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_4200_dvd_Oless__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).

tff(fact_4201_dvd_Oless__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_4202_dvd_Oless__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( X != Y ) ) ).

tff(fact_4203_dvd_Oless__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( Y != X ) ) ).

tff(fact_4204_dvd_Oless__imp__triv,axiom,
    ! [P_1: bool,X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) )
       => hBOOL(P_1) ) ) ).

tff(fact_4205_dvd_Oord__less__eq__trans,axiom,
    ! [C_1: nat,A_2: nat,B_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A_2)) )
     => ( ( B_1 = C_1 )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),C_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_2)) ) ) ) ).

tff(fact_4206_dvd_Oless__le__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).

tff(fact_4207_dvd_Oless__asym_H,axiom,
    ! [A_2: nat,B_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A_2)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A_2))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1)) ) ) ).

tff(fact_4208_dvd_Oless__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).

tff(fact_4209_dvd_Oless__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_4210_divides__antisym,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_4211_dvd__diff__nat,axiom,
    ! [N_1: nat,K_2: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1))) ) ) ).

tff(fact_4212_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> ( M = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

tff(fact_4213_nat__dvd__not__less,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
       => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M_1)) ) ) ).

tff(fact_4214_dvd__pos__nat,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1)) ) ) ).

tff(fact_4215_divides__ge,axiom,
    ! [A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1))
     => ( ( B_1 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),B_1)) ) ) ).

tff(fact_4216_divides__le,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_4217_nat__mult__dvd__cancel__disj_H,axiom,
    ! [M: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)))
    <=> ( ( K_1 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) ) ) ).

tff(fact_4218_nat__mult__dvd__cancel__disj,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
    <=> ( ( K_1 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) ) ) ).

tff(fact_4219_dvd__diffD1,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) ) ) ) ).

tff(fact_4220_dvd__diffD,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M_1)) ) ) ) ).

tff(fact_4221_divides__rev,axiom,
    ! [A_2: nat,N_1: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),N_1)))
     => ( ( N_1 != zero_zero_nat )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_1)) ) ) ).

tff(fact_4222_divides__exp2,axiom,
    ! [X: nat,Y: nat,N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1)),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_4223_pow__divides__eq__nat,axiom,
    ! [Aa: nat,Ba: nat,N: nat] :
      ( ( N != zero_zero_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Aa),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ba),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Aa),Ba)) ) ) ).

tff(fact_4224_divides__rexp,axiom,
    ! [N_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_nat_nat(suc,N_1)))) ) ).

tff(fact_4225_fact__dvd,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N_1)),hAPP_nat_nat(fact_fact_nat,M_1))) ) ).

tff(fact_4226_divides__mul,axiom,
    ! [N_1: nat,M_1: nat,R_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),R_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),R_3))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,M_1),N_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),R_3)) ) ) ) ).

tff(fact_4227_coprime__divprod,axiom,
    ! [D_3: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_1)) ) ) ).

tff(fact_4228_coprime,axiom,
    ! [Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Aa),Ba))
    <=> ! [D_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),Aa))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),Ba)) )
        <=> ( D_2 = one_one_nat ) ) ) ).

tff(fact_4229_dvd__imp__le,axiom,
    ! [K_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1)) ) ) ).

tff(fact_4230_dvd__mult__cancel,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),N_1)) ) ) ).

tff(fact_4231_nat__mult__dvd__cancel1,axiom,
    ! [M: nat,N: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) ) ) ).

tff(fact_4232_dvd__fact__nat,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),hAPP_nat_nat(fact_fact_nat,N_1))) ) ) ).

tff(fact_4233_divides__div__not,axiom,
    ! [X: nat,Q_1: nat,N_1: nat,R_3: nat] :
      ( ( X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_1),N_1)),R_3) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_3))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),N_1))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),X)) ) ) ) ).

tff(fact_4234_dvd__mult__cancel2,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

tff(fact_4235_dvd__mult__cancel1,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

tff(fact_4236_power__dvd__imp__le,axiom,
    ! [I: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N_1)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ) ).

tff(fact_4237_even__dvd,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)) ) ).

tff(fact_4238_nat__even__iff__2__dvd,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) ) ).

tff(fact_4239_real__of__nat__div,axiom,
    ! [N_1: nat,D_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),N_1))
       => ( hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,D_3)) ) ) ) ).

tff(fact_4240_dvd_Olift__Suc__mono__le,axiom,
    ! [N: nat,N_5: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N_5))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,N_5))) ) ) ).

tff(fact_4241_dvd_Olift__Suc__mono__less__iff,axiom,
    ! [N: nat,M: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,M)))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,M)),hAPP_nat_nat(F,N))) )
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) ) ) ).

tff(fact_4242_dvd_Olift__Suc__mono__less,axiom,
    ! [N: nat,N_5: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N)),hAPP_nat_nat(F,N_5)))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_5)),hAPP_nat_nat(F,N))) ) ) ) ).

tff(fact_4243_zdvd__self__abs2,axiom,
    ! [D_3: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,D_3)),D_3)) ).

tff(fact_4244_zdvd__self__abs1,axiom,
    ! [D_3: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),hAPP_int_int(abs_abs_int,D_3))) ).

tff(fact_4245_zdvd__zmod,axiom,
    ! [N_1: int,F_3: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M_1),N_1))) ) ) ).

tff(fact_4246_zdvd__zmod__imp__zdvd,axiom,
    ! [K_2: int,M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(div_mod_int(M_1),N_1)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1)) ) ) ).

tff(fact_4247_zdvd__antisym__abs,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),B_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_1),A_2))
       => ( hAPP_int_int(abs_abs_int,A_2) = hAPP_int_int(abs_abs_int,B_1) ) ) ) ).

tff(fact_4248_zdvd__zdiffD,axiom,
    ! [K_2: int,M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M_1),N_1)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1)) ) ) ).

tff(fact_4249_uminus__dvd__conv_I1_J,axiom,
    ! [D: int,T: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),T))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D)),T)) ) ).

tff(fact_4250_uminus__dvd__conv_I2_J,axiom,
    ! [D: int,T: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),T))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(uminus_uminus_int,T))) ) ).

tff(fact_4251_zdvd__not__zless,axiom,
    ! [N_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M_1),N_1))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M_1)) ) ) ).

tff(fact_4252_zdvd__antisym__nonneg,axiom,
    ! [N_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),N_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M_1))
           => ( M_1 = N_1 ) ) ) ) ) ).

tff(fact_4253_zdvd__bounds,axiom,
    ! [N_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_1),zero_zero_int))
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),M_1)) ) ) ).

tff(fact_4254_zdvd__mono,axiom,
    ! [M: int,T: int,K_1: int] :
      ( ( K_1 != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),T))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),T))) ) ) ).

tff(fact_4255_zdvd__mult__cancel,axiom,
    ! [K_2: int,M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)))
     => ( ( K_2 != zero_zero_int )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),N_1)) ) ) ).

tff(fact_4256_zdvd__period,axiom,
    ! [C: int,X_1: int,T: int,Aa: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Aa),D))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),T)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D))),T))) ) ) ).

tff(fact_4257_zdvd__reduce,axiom,
    ! [K_1: int,N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),M))))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) ) ).

tff(fact_4258_zcong__zero__equiv__div,axiom,
    ! [Aa: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),M))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),Aa)) ) ).

tff(fact_4259_zcong__eq__zdvd__prop,axiom,
    ! [X_1: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),Pa))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Pa),X_1)) ) ).

tff(fact_4260_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) ) ).

tff(fact_4261_zdvd__int,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))) ) ).

tff(fact_4262_pow__divides__pow__int,axiom,
    ! [A_2: int,N_1: nat,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),N_1)))
     => ( ( N_1 != zero_zero_nat )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),B_1)) ) ) ).

tff(fact_4263_pow__divides__eq__int,axiom,
    ! [Aa: int,Ba: int,N: nat] :
      ( ( N != zero_zero_nat )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Aa),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Ba),N)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Aa),Ba)) ) ) ).

tff(fact_4264_zdvd__mult__div__cancel,axiom,
    ! [N_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M_1),N_1)) = M_1 ) ) ).

tff(fact_4265_zdvd1__eq,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),one_one_int))
    <=> ( hAPP_int_int(abs_abs_int,X_1) = one_one_int ) ) ).

tff(fact_4266_zcong__def,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba))) ) ).

tff(fact_4267_zmod__eq__dvd__iff,axiom,
    ! [X_1: int,N: int,Y_2: int] :
      ( ( hAPP_int_int(div_mod_int(X_1),N) = hAPP_int_int(div_mod_int(Y_2),N) )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2))) ) ).

tff(fact_4268_zprime__zdvd__zmult__better,axiom,
    ! [M_1: int,N_1: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M_1))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) ) ) ) ).

tff(fact_4269_abs__div,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X))
     => ( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) ) ) ).

tff(fact_4270_zprime__zdvd__power,axiom,
    ! [A_2: int,N_1: nat,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_2)) ) ) ).

tff(fact_4271_zdvd__imp__le,axiom,
    ! [Z_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N_1)) ) ) ).

tff(fact_4272_zpower__zdvd__prop1,axiom,
    ! [P_3: int,Y: int,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N_1))) ) ) ).

tff(fact_4273_zmod__zdvd__zmod,axiom,
    ! [A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),B_1))
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_2),B_1)),M_1) = hAPP_int_int(div_mod_int(A_2),M_1) ) ) ) ).

tff(fact_4274_zdvd__iff__zmod__eq__0__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_1)) = zero_zero_int ) ) ).

tff(fact_4275_dvd__imp__le__int,axiom,
    ! [D_3: int,I: int] :
      ( ( I != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),I))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,D_3)),hAPP_int_int(abs_abs_int,I))) ) ) ).

tff(fact_4276_zprime__power__zdvd__cancel__right,axiom,
    ! [N_1: nat,A_2: int,B_1: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),B_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),A_2)) ) ) ) ).

tff(fact_4277_zprime__power__zdvd__cancel__left,axiom,
    ! [N_1: nat,B_1: int,A_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N_1)),B_1)) ) ) ) ).

tff(fact_4278_transfer__nat__int__relations_I4_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat_1,X_1)),hAPP_int_nat(nat_1,Y_2)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),Y_2)) ) ) ) ).

tff(fact_4279_zdvd__mult__cancel1,axiom,
    ! [N: int,M: int] :
      ( ( M != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)),M))
      <=> ( hAPP_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

tff(fact_4280_zprime__zdvd__zmult,axiom,
    ! [N_1: int,P_3: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M_1))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1)) ) ) ) ) ).

tff(fact_4281_real__of__int__div,axiom,
    ! [N_1: int,D_3: int] :
      ( ( D_3 != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N_1))
       => ( hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N_1)),hAPP_int_real(real_int,D_3)) ) ) ) ).

tff(fact_4282_dvd__int__iff,axiom,
    ! [Z_2: int,M: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Z_2))),M)) ) ).

tff(fact_4283_int__dvd__iff,axiom,
    ! [M: nat,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,M)),Z_2))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Z_2)))) ) ).

tff(fact_4284_zpower__zdvd__prop2,axiom,
    ! [Y: int,N_1: nat,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N_1)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ) ).

tff(fact_4285_zprime__def,axiom,
    ! [Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Pa))
        & ! [M_2: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_2))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_2),Pa)) )
           => ( ( M_2 = one_one_int )
              | ( M_2 = Pa ) ) ) ) ) ).

tff(fact_4286_int__even__iff__2__dvd,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) ) ).

tff(fact_4287_nat__dvd__iff,axiom,
    ! [Z_2: int,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat_1,Z_2)),M))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,M))) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
         => ( M = zero_zero_nat ) ) ) ) ).

tff(fact_4288_Little__Fermat,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))),one_one_int),P_3)) ) ) ).

tff(fact_4289_Euler_Oaux____1,axiom,
    ! [Y: int,X: int,P_3: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),P_3))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ).

tff(fact_4290_zprime__factor__exists,axiom,
    ! [A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
     => ? [P_4: int] :
          ( hBOOL(hAPP_int_bool(zprime,P_4))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_2)) ) ) ).

tff(fact_4291_prime__dvd__power__two,axiom,
    ! [M_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M_1)) ) ) ).

tff(fact_4292_one__code__int__code,axiom,
    one_on1684967323de_int = hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit1,pls)) ).

tff(fact_4293_Quickcheck__Narrowing_Oint__of__code,axiom,
    ! [K_2: quickcheck_code_int] :
      ( ( ( K_2 = zero_z891286103de_int )
       => ( quickcheck_int_of(K_2) = zero_zero_int ) )
      & ( ( K_2 != zero_z891286103de_int )
       => ( ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int )
           => ( quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) )
          & ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z891286103de_int )
           => ( quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) ) ) ) ) ) ).

tff(fact_4294_distinct__prime__coprime,axiom,
    ! [Q_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(prime,Q_1))
       => ( ( P_3 != Q_1 )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_1)) ) ) ) ).

tff(fact_4295_less__eq__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,N),M))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ) ).

tff(fact_4296_less__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,N),M))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ) ).

tff(fact_4297_prime__1,axiom,
    ~ hBOOL(hAPP_nat_bool(prime,one_one_nat)) ).

tff(fact_4298_prime__0,axiom,
    ~ hBOOL(hAPP_nat_bool(prime,zero_zero_nat)) ).

tff(fact_4299_int__of__number,axiom,
    ! [K_2: int] : quickcheck_int_of(hAPP_i1732201573de_int(number1226105091de_int,K_2)) = hAPP_int_int(number_number_of_int,K_2) ).

tff(fact_4300_code__int_Oint__of__inject,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ( quickcheck_int_of(X_1) = quickcheck_int_of(Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_4301_Quickcheck__Narrowing_Oint__of__inject,axiom,
    ! [K_1: quickcheck_code_int,L: quickcheck_code_int] :
      ( ( quickcheck_int_of(K_1) = quickcheck_int_of(L) )
    <=> ( K_1 = L ) ) ).

tff(fact_4302_primes__eq,axiom,
    ! [Q_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(prime,Q_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_1))
         => ( P_3 = Q_1 ) ) ) ) ).

tff(fact_4303_prime__Suc0,axiom,
    ~ hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_4304_prime__g__zero,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),P_3)) ) ).

tff(fact_4305_prime__divprod__eq,axiom,
    ! [Aa: nat,Ba: nat,Pa: nat] :
      ( hBOOL(hAPP_nat_bool(prime,Pa))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Pa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),Ba)))
      <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Pa),Aa))
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Pa),Ba)) ) ) ) ).

tff(fact_4306_prime__divprod,axiom,
    ! [A_2: nat,B_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_2))
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_1)) ) ) ) ).

tff(fact_4307_prime__dvd__mult,axiom,
    ! [M_1: nat,N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M_1))
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1)) ) ) ) ).

tff(fact_4308_prime__product,axiom,
    ! [P_3: nat,Q_1: nat] :
      ( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_1)))
     => ( ( P_3 = one_one_nat )
        | ( Q_1 = one_one_nat ) ) ) ).

tff(fact_4309_prime__dvd__power,axiom,
    ! [A_2: nat,N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_2)) ) ) ).

tff(fact_4310_prime__divexp,axiom,
    ! [X: nat,N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X)) ) ) ).

tff(fact_4311_prime__divexp__n,axiom,
    ! [X: nat,N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1))) ) ) ).

tff(fact_4312_prime__exp,axiom,
    ! [Pa: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Pa),N)))
    <=> ( hBOOL(hAPP_nat_bool(prime,Pa))
        & ( N = one_one_nat ) ) ) ).

tff(fact_4313_coprime__prime,axiom,
    ! [P_3: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),B_1))
     => ~ ( hBOOL(hAPP_nat_bool(prime,P_3))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_2))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_1)) ) ) ).

tff(fact_4314_prime__coprime__strong,axiom,
    ! [N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N_1)) ) ) ).

tff(fact_4315_coprime__prime__eq,axiom,
    ! [Aa: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Aa),Ba))
    <=> ! [P_4: nat] :
          ~ ( hBOOL(hAPP_nat_bool(prime,P_4))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Aa))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Ba)) ) ) ).

tff(fact_4316_prime__impl__zprime__int,axiom,
    ! [A_2: nat] :
      ( hBOOL(hAPP_nat_bool(prime,A_2))
     => hBOOL(hAPP_int_bool(zprime,hAPP_nat_int(semiri1621563631at_int,A_2))) ) ).

tff(fact_4317_zero__code__int__code,axiom,
    zero_z891286103de_int = hAPP_i1732201573de_int(number1226105091de_int,pls) ).

tff(fact_4318_prime__g__one,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),P_3)) ) ).

tff(fact_4319_prime__nd__one,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(suc,zero_zero_nat))) ) ).

tff(fact_4320_prime__factor__lt,axiom,
    ! [M_1: nat,N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( N_1 != zero_zero_nat )
       => ( ( N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),M_1) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ) ).

tff(fact_4321_prime__def,axiom,
    ! [Pa: nat] :
      ( hBOOL(hAPP_nat_bool(prime,Pa))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),Pa))
        & ! [M_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_2),Pa))
           => ( ( M_2 = one_one_nat )
              | ( M_2 = Pa ) ) ) ) ) ).

tff(fact_4322_divides__primepow,axiom,
    ! [D: nat,K_1: nat,Pa: nat] :
      ( hBOOL(hAPP_nat_bool(prime,Pa))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Pa),K_1)))
      <=> ? [I_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1))
            & ( D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Pa),I_2) ) ) ) ) ).

tff(fact_4323_prime__power__dvd__cancel__right,axiom,
    ! [N_1: nat,A_2: nat,B_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),A_2)) ) ) ) ).

tff(fact_4324_prime__coprime__lt,axiom,
    ! [X: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),P_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),P_3)) ) ) ) ).

tff(fact_4325_prime__coprime,axiom,
    ! [N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( N_1 = one_one_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N_1))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N_1)) ) ) ).

tff(fact_4326_two__is__prime,axiom,
    hBOOL(hAPP_nat_bool(prime,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_4327_prime__dvd__square,axiom,
    ! [M_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M_1)) ) ) ).

tff(fact_4328_prime__divprod__pow,axiom,
    ! [N_1: nat,A_2: nat,B_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),B_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),A_2))
            | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N_1)),B_1)) ) ) ) ) ).

tff(fact_4329_prime__ge__2,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) ) ).

tff(fact_4330_prime__odd,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( P_3 = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) )
        | ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)) ) ) ).

tff(fact_4331_div__mod__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,M_1: quickcheck_code_int] : quickc495462417de_int(N_1,M_1) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N_1),M_1)),hAPP_Q1762011733de_int(div_mo231679042de_int(N_1),M_1)) ).

tff(fact_4332_nat__of__def,axiom,
    ! [I: quickcheck_code_int] : quickcheck_nat_of(I) = hAPP_int_nat(nat_1,quickcheck_int_of(I)) ).

tff(fact_4333_coprime__prime__dvd__ex,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y))
     => ? [P_4: nat] :
          ( hBOOL(hAPP_nat_bool(prime,P_4))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Y)) ) ) ).

tff(fact_4334_prime__power__exp,axiom,
    ! [X: nat,K_2: nat,N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( N_1 != zero_zero_nat )
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) )
         => ? [I_2: nat] : X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) ) ) ) ).

tff(fact_4335_prime__factor,axiom,
    ! [N_1: nat] :
      ( ( N_1 != one_one_nat )
     => ? [P_4: nat] :
          ( hBOOL(hAPP_nat_bool(prime,P_4))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),N_1)) ) ) ).

tff(fact_4336_bezout__prime,axiom,
    ! [A_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_2))
       => ? [X_2: nat,Y_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Y_1)),one_one_nat) ) ) ).

tff(fact_4337_not__prime__ex__mk,axiom,
    ! [N_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
        & ~ hBOOL(hAPP_nat_bool(prime,N_1)) )
     => ? [M_2: nat,K: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_2))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N_1))
          & ( N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),K) ) ) ) ).

tff(fact_4338_euclid,axiom,
    ! [N_1: nat] :
    ? [P_4: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_4))
      & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),P_4)) ) ).

tff(fact_4339_around__zero_Opinduct,axiom,
    ! [P_1: fun_int_bool,A0: int] :
      ( hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),A0))
     => ( ! [I_2: int] :
            ( hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),I_2))
           => ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),zero_zero_int))
               => ( ( I_2 != zero_zero_int )
                 => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
             => hBOOL(hAPP_int_bool(P_1,I_2)) ) )
       => hBOOL(hAPP_int_bool(P_1,A0)) ) ) ).

tff(fact_4340_DeMoivre2,axiom,
    ! [R_3: real,A_2: real,N_1: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_3,A_2)),N_1) = rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),N_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),A_2)) ).

tff(fact_4341_fact__int__def,axiom,
    ! [X: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
       => ( hAPP_int_int(fact_fact_int,X) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat_1,X))) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
       => ( hAPP_int_int(fact_fact_int,X) = zero_zero_int ) ) ) ).

tff(fact_4342_fact__mono__int,axiom,
    ! [M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_1),N_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M_1)),hAPP_int_int(fact_fact_int,N_1))) ) ).

tff(fact_4343_fact__1__int,axiom,
    hAPP_int_int(fact_fact_int,one_one_int) = one_one_int ).

tff(fact_4344_fact__neg__int,axiom,
    ! [M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M_1),zero_zero_int))
     => ( hAPP_int_int(fact_fact_int,M_1) = zero_zero_int ) ) ).

tff(fact_4345_fact__less__mono__int,axiom,
    ! [N_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M_1),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M_1)),hAPP_int_int(fact_fact_int,N_1))) ) ) ).

tff(fact_4346_fact__ge__zero__int,axiom,
    ! [M_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,M_1))) ).

tff(fact_4347_fact__nonzero__int,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_int_int(fact_fact_int,N_1) != zero_zero_int ) ) ).

tff(fact_4348_transfer__nat__int__factorial__closure,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))) ) ).

tff(fact_4349_fact__0__int,axiom,
    hAPP_int_int(fact_fact_int,zero_zero_int) = one_one_int ).

tff(fact_4350_transfer__int__nat__factorial,axiom,
    ! [X: nat] : hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X)) ).

tff(fact_4351_rcis__zero__arg,axiom,
    ! [R_3: real] : rcis(R_3,zero_zero_real) = of_real_complex(R_3) ).

tff(fact_4352_complex__mod__rcis,axiom,
    ! [R_3: real,A_2: real] : hAPP_complex_real(norm_norm_complex,rcis(R_3,A_2)) = hAPP_real_real(abs_abs_real,R_3) ).

tff(fact_4353_cis__rcis__eq,axiom,
    ! [A_2: real] : cis(A_2) = rcis(one_one_real,A_2) ).

tff(fact_4354_rcis__zero__mod,axiom,
    ! [A_2: real] : rcis(zero_zero_real,A_2) = zero_zero_complex ).

tff(fact_4355_fact__gt__zero__int,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(fact_fact_int,N_1))) ) ).

tff(fact_4356_fact__ge__one__int,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(fact_fact_int,N_1))) ) ).

tff(fact_4357_fact__mono__int__aux,axiom,
    ! [M_1: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M_1)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_1),K_2)))) ) ).

tff(fact_4358_dvd__fact__int,axiom,
    ! [N_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_1),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),hAPP_int_int(fact_fact_int,N_1))) ) ) ).

tff(fact_4359_Re__rcis,axiom,
    ! [R_3: real,A_2: real] : hAPP_complex_real(re,rcis(R_3,A_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_2)) ).

tff(fact_4360_Im__rcis,axiom,
    ! [R_3: real,A_2: real] : hAPP_complex_real(im,rcis(R_3,A_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_2)) ).

tff(fact_4361_rcis__mult,axiom,
    ! [R1: real,A_2: real,R2: real,B_1: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,rcis(R1,A_2)),rcis(R2,B_1)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_2),B_1)) ).

tff(fact_4362_transfer__nat__int__factorial,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat_1,X)) = hAPP_int_nat(nat_1,hAPP_int_int(fact_fact_int,X)) ) ) ).

tff(fact_4363_rcis__divide,axiom,
    ! [R1: real,A_2: real,R2: real,B_1: real] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,rcis(R1,A_2)),rcis(R2,B_1)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_2),B_1)) ).

tff(fact_4364_rcis__def,axiom,
    ! [R_3: real,A_2: real] : rcis(R_3,A_2) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),cis(A_2)) ).

tff(fact_4365_fact__less__mono__int__aux,axiom,
    ! [M_1: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M_1)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_1),one_one_int)),K_2)))) ) ) ).

tff(fact_4366_fact__plus__one__int,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)),hAPP_int_int(fact_fact_int,N_1)) ) ) ).

tff(fact_4367_fact__reduce__int,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( hAPP_int_int(fact_fact_int,N_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int))) ) ) ).

tff(fact_4368_rcis__inverse,axiom,
    ! [R_3: real,A_2: real] : hAPP_complex_complex(invers1449016382omplex,rcis(R_3,A_2)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_3),hAPP_real_real(uminus_uminus_real,A_2)) ).

tff(fact_4369_length__around__zero,axiom,
    ! [I: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I))
     => ( hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat_1,I))),one_one_nat) ) ) ).

tff(fact_4370_Bnor__prime,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Pa))
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(Aa),Pa)) = hAPP_int_nat(nat_1,Aa) ) ) ) ).

tff(fact_4371_Bnor__mem__zle,axiom,
    ! [Ba: int,Aa: int,M: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(bnorRset(Aa),M)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa)) ) ).

tff(fact_4372_Bnor__mem__zle__swap,axiom,
    ! [M: int,Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(bnorRset(Aa),M))) ) ).

tff(fact_4373_Bnor__fin,axiom,
    ! [Aa: int,M: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(Aa),M))) ).

tff(fact_4374_Bnor__mem__zg,axiom,
    ! [Ba: int,Aa: int,M: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(bnorRset(Aa),M)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba)) ) ).

tff(fact_4375_lemma__BOLZANO,axiom,
    ! [Aa: real,Ba: real,P_1: fun_Pr1127657839l_bool] :
      ( ! [A_1: real,B_3: real,C_3: real] :
          ( ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3)))
            & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_3),C_3)))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_3))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),C_3)) )
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),C_3))) )
     => ( ! [X_2: real] :
          ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [A_1: real,B_3: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_2))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B_3))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_3),A_1)),D_2)) )
               => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3))) ) )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,Aa),Ba))) ) ) ) ).

tff(fact_4376_lemma__BOLZANO2,axiom,
    ! [P_1: fun_Pr1127657839l_bool] :
      ( ( ! [A_1: real,B_3: real,C_3: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_3))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),C_3))
              & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3)))
              & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_3),C_3))) )
           => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),C_3))) )
        & ! [X_2: real] :
          ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [A_1: real,B_3: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_2))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B_3))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_3),A_1)),D_2)) )
               => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3))) ) ) )
     => ! [A_1: real,B_3: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_3))
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_3))) ) ) ).

tff(fact_4377_bezw__aux,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,Y))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).

tff(fact_4378_gcd__dvd2__nat,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)),N_1)) ).

tff(fact_4379_gcd__dvd1__nat,axiom,
    ! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)),M_1)) ).

tff(fact_4380_gcd__dvd__prod__nat,axiom,
    ! [M_1: nat,N_1: nat,K_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1))) ).

tff(fact_4381_coprime__exp2__nat,axiom,
    ! [N_1: nat,M_1: nat,A_2: nat,B_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),M_1)) = one_one_nat ) ) ).

tff(fact_4382_gcd__le2__nat,axiom,
    ! [A_2: nat,B_1: nat] :
      ( ( B_1 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),B_1)) ) ).

tff(fact_4383_gcd__le1__nat,axiom,
    ! [B_1: nat,A_2: nat] :
      ( ( A_2 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),A_2)) ) ).

tff(fact_4384_gcd__Suc__0,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_4385_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

tff(fact_4386_gcd__exp__nat,axiom,
    ! [A_2: nat,N_1: nat,B_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),N_1) ).

tff(fact_4387_gcd__semilattice__nat_Oinf_Oidem,axiom,
    ! [A_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),A_2) = A_2 ).

tff(fact_4388_gcd__idem__nat,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).

tff(fact_4389_gcd__commute__nat,axiom,
    ! [A_2: nat,B_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_1),A_2) ).

tff(fact_4390_gcd__semilattice__nat_Oinf__commute,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),X) ).

tff(fact_4391_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
    ! [A_2: nat,B_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1) ).

tff(fact_4392_gcd__semilattice__nat_Oinf__left__idem,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) ).

tff(fact_4393_gcd__left__commute__nat,axiom,
    ! [B_1: nat,A_2: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_1),C_1)) ).

tff(fact_4394_gcd__semilattice__nat_Oinf__left__commute,axiom,
    ! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Z_1)) ).

tff(fact_4395_gcd__assoc__nat,axiom,
    ! [A_2: nat,B_1: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_1),C_1)) ).

tff(fact_4396_gcd__semilattice__nat_Oinf__assoc,axiom,
    ! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) ).

tff(fact_4397_gcd__mult__distrib__nat,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_1)) ).

tff(fact_4398_gcd__add1__nat,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) ).

tff(fact_4399_gcd__add2__nat,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) ).

tff(fact_4400_gcd__semilattice__nat_Oless__infI2,axiom,
    ! [A_2: nat,B_1: nat,X: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_1)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1))) ) ) ).

tff(fact_4401_gcd__semilattice__nat_Oless__infI1,axiom,
    ! [B_1: nat,A_2: nat,X: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_2)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1))) ) ) ).

tff(fact_4402_gcd__semilattice__nat_Ole__infE,axiom,
    ! [X: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_2))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_1)) ) ) ).

tff(fact_4403_dvd__gcd__D2__nat,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1)) ) ).

tff(fact_4404_dvd__gcd__D1__nat,axiom,
    ! [K_2: nat,M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M_1)) ) ).

tff(fact_4405_gcd__semilattice__nat_Oinf__mono,axiom,
    ! [B_1: nat,D_3: nat,A_2: nat,C_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),C_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),D_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,C_1),D_3))) ) ) ).

tff(fact_4406_gcd__semilattice__nat_Oinf__greatest,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1))) ) ) ).

tff(fact_4407_gcd__semilattice__nat_Ole__infI,axiom,
    ! [B_1: nat,X: nat,A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1))) ) ) ).

tff(fact_4408_gcd__greatest__nat,axiom,
    ! [N_1: nat,K_2: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1))) ) ) ).

tff(fact_4409_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ) ).

tff(fact_4410_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ) ).

tff(fact_4411_gcd__semilattice__nat_Ole__infI2,axiom,
    ! [A_2: nat,B_1: nat,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),X))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),X)) ) ).

tff(fact_4412_gcd__semilattice__nat_Ole__infI1,axiom,
    ! [B_1: nat,A_2: nat,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),X))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)),X)) ) ).

tff(fact_4413_gcd__unique__nat,axiom,
    ! [Ba: nat,D: nat,Aa: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),Aa))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),Ba))
        & ! [E: nat] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),Aa))
              & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),Ba)) )
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D)) ) )
    <=> ( D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) ) ) ).

tff(fact_4414_gcd__semilattice__nat_Ole__inf__iff,axiom,
    ! [X_1: nat,Y_2: nat,Z_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_2),Z_2)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_2)) ) ) ).

tff(fact_4415_gcd__greatest__iff__nat,axiom,
    ! [K_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) ) ) ).

tff(fact_4416_gcd__semilattice__nat_Ole__iff__inf,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_2) = X_1 ) ) ).

tff(fact_4417_gcd__semilattice__nat_Oinf__le2,axiom,
    ! [X: nat,Y: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Y)) ).

tff(fact_4418_gcd__semilattice__nat_Oinf__le1,axiom,
    ! [X: nat,Y: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),X)) ).

tff(fact_4419_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X) = one_one_nat ).

tff(fact_4420_gcd__1__nat,axiom,
    ! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),one_one_nat) = one_one_nat ).

tff(fact_4421_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),one_one_nat) = one_one_nat ).

tff(fact_4422_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_2) = zero_zero_nat )
    <=> ( ( X_1 = zero_zero_nat )
        & ( Y_2 = zero_zero_nat ) ) ) ).

tff(fact_4423_gcd__zero__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

tff(fact_4424_gcd__0__nat,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

tff(fact_4425_gcd__0__left__nat,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

tff(fact_4426_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ).

tff(fact_4427_coprime__Suc__nat,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N_1)),N_1) = one_one_nat ).

tff(fact_4428_gcd__add__mult__nat,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) ).

tff(fact_4429_coprime__common__divisor__nat,axiom,
    ! [X: nat,A_2: nat,B_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1) = one_one_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_1))
         => ( X = one_one_nat ) ) ) ) ).

tff(fact_4430_coprime__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) = one_one_nat )
    <=> ! [D_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),Aa))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),Ba)) )
        <=> ( D_2 = one_one_nat ) ) ) ).

tff(fact_4431_coprime__plus__one__nat,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),N_1) = one_one_nat ).

tff(fact_4432_coprime__rmult__nat,axiom,
    ! [D_3: nat,A_2: nat,B_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_1) = one_one_nat ) ) ).

tff(fact_4433_coprime__lmult__nat,axiom,
    ! [D_3: nat,A_2: nat,B_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_2) = one_one_nat ) ) ).

tff(fact_4434_coprime__crossproduct__nat,axiom,
    ! [Ba: nat,C: nat,Aa: nat,D: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Aa),D) = one_one_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ba),C) = one_one_nat )
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),D) )
        <=> ( ( Aa = Ba )
            & ( C = D ) ) ) ) ) ).

tff(fact_4435_coprime__mult__nat,axiom,
    ! [B_1: nat,D_3: nat,A_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_2) = one_one_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_1) = one_one_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1)) = one_one_nat ) ) ) ).

tff(fact_4436_gcd__mult__cancel__nat,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N_1) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) ) ) ).

tff(fact_4437_coprime__mul__eq__nat,axiom,
    ! [D: nat,Aa: nat,Ba: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Aa),Ba)) = one_one_nat )
    <=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),Aa) = one_one_nat )
        & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),Ba) = one_one_nat ) ) ) ).

tff(fact_4438_gcd__diff1__nat,axiom,
    ! [N_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) ) ) ).

tff(fact_4439_gcd__diff2__nat,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) ) ) ).

tff(fact_4440_gcd__nat_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ) ).

tff(fact_4441_gcd__non__0__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( Y != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ).

tff(fact_4442_coprime__exp__nat,axiom,
    ! [N_1: nat,D_3: nat,A_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_2) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_2),N_1)) = one_one_nat ) ) ).

tff(fact_4443_gcd__coprime__nat,axiom,
    ! [B_4: nat,A_4: nat,A_2: nat,B_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1) != zero_zero_nat )
     => ( ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)) )
       => ( ( B_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)) )
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_4),B_4) = one_one_nat ) ) ) ) ).

tff(fact_4444_coprime__minus__one__nat,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),N_1) = one_one_nat ) ) ).

tff(fact_4445_divides__mult__nat,axiom,
    ! [N_1: nat,M_1: nat,R_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),R_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),R_3))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) = one_one_nat )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),R_3)) ) ) ) ).

tff(fact_4446_coprime__dvd__mult__iff__nat,axiom,
    ! [M: nat,K_1: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_1),N) = one_one_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)) ) ) ).

tff(fact_4447_coprime__dvd__mult__nat,axiom,
    ! [M_1: nat,K_2: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N_1) = one_one_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M_1)) ) ) ).

tff(fact_4448_div__gcd__coprime__nat,axiom,
    ! [B_1: nat,A_2: nat] :
      ( ( ( A_2 != zero_zero_nat )
        | ( B_1 != zero_zero_nat ) )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1))) = one_one_nat ) ) ).

tff(fact_4449_invertible__coprime__nat,axiom,
    ! [X: nat,Y: nat,M_1: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),M_1) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),M_1) = one_one_nat ) ) ).

tff(fact_4450_coprime__Suc__0__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) = one_one_nat )
    <=> ! [D_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),Aa))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),Ba)) )
        <=> ( D_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_4451_gcd__coprime__exists__nat,axiom,
    ! [A_2: nat,B_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1) != zero_zero_nat )
     => ? [A_3: nat,B_2: nat] :
          ( ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)) )
          & ( B_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)) )
          & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat ) ) ) ).

tff(fact_4452_bezout__nat,axiom,
    ! [B_1: nat,A_2: nat] :
      ( ( A_2 != zero_zero_nat )
     => ? [X_2: nat,Y_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_2),B_1)) ) ).

tff(fact_4453_Cauchy__iff2,axiom,
    ! [X_3: fun_nat_real] :
      ( hBOOL(cauchy_real(X_3))
    <=> ! [J_1: nat] :
        ? [M_4: nat] :
        ! [M_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_4),M_2))
         => ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_4),N_2))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(X_3,M_2)),hAPP_nat_real(X_3,N_2)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))) ) ) ) ).

tff(fact_4454_ln__powr__bound,axiom,
    ! [A_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),A_2))) ) ) ).

tff(fact_4455_ln__powr__bound2,axiom,
    ! [A_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(ln,X)),A_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_2),A_2)),X))) ) ) ).

tff(fact_4456_powr__one__eq__one,axiom,
    ! [A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_2) = one_one_real ).

tff(fact_4457_powr__not__zero,axiom,
    ! [X: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2) != zero_zero_real ).

tff(fact_4458_powr__powr__swap,axiom,
    ! [X: real,A_2: real,B_1: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),B_1) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_1)),A_2) ).

tff(fact_4459_powr__powr,axiom,
    ! [X: real,A_2: real,B_1: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),B_1) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),B_1)) ).

tff(fact_4460_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y))) ).

tff(fact_4461_powr__less__mono2__neg,axiom,
    ! [Y: real,X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2))) ) ) ) ).

tff(fact_4462_powr__less__mono2,axiom,
    ! [Y: real,X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_2))) ) ) ) ).

tff(fact_4463_powr__gt__zero,axiom,
    ! [X: real,A_2: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2))) ).

tff(fact_4464_powr__mono,axiom,
    ! [X: real,A_2: real,B_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_1))) ) ) ).

tff(fact_4465_powr__zero__eq__one,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),zero_zero_real) = one_one_real ).

tff(fact_4466_powr__less__cancel,axiom,
    ! [X: real,A_2: real,B_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_1)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_1)) ) ) ).

tff(fact_4467_powr__less__mono,axiom,
    ! [X: real,A_2: real,B_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_1))) ) ) ).

tff(fact_4468_powr__less__cancel__iff,axiom,
    ! [Aa: real,Ba: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Aa)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Ba)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ) ).

tff(fact_4469_powr__add,axiom,
    ! [X: real,A_2: real,B_1: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_2),B_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_1)) ).

tff(fact_4470_powr__divide2,axiom,
    ! [X: real,A_2: real,B_1: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_1)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_2),B_1)) ).

tff(fact_4471_powr__minus,axiom,
    ! [X: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_2)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)) ).

tff(fact_4472_powr__mono2,axiom,
    ! [Y: real,X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_2))) ) ) ) ).

tff(fact_4473_powr__mult,axiom,
    ! [A_2: real,Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)),A_2) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_2)) ) ) ) ).

tff(fact_4474_ge__one__powr__ge__zero,axiom,
    ! [A_2: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2))) ) ) ).

tff(fact_4475_powr__le__cancel__iff,axiom,
    ! [Aa: real,Ba: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Aa)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),Ba)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ) ).

tff(fact_4476_powr__one__gt__zero__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),one_one_real) = X_1 )
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_4477_powr__divide,axiom,
    ! [A_2: real,Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)),A_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_2)) ) ) ) ).

tff(fact_4478_powr__minus__divide,axiom,
    ! [X: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_2)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2)) ).

tff(fact_4479_powr__def,axiom,
    ! [X: real,A_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_2) = hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),hAPP_real_real(ln,X))) ).

tff(fact_4480_powr__realpow,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N_1)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1) ) ) ).

tff(fact_4481_ln__powr,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(ln,X)) ) ) ) ).

tff(fact_4482_log__powr__cancel,axiom,
    ! [Y: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hAPP_real_real(log(A_2),hAPP_real_real(hAPP_r1250527377l_real(powr,A_2),Y)) = Y ) ) ) ).

tff(fact_4483_powr__log__cancel,axiom,
    ! [X: real,A_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_2))
     => ( ( A_2 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hAPP_real_real(hAPP_r1250527377l_real(powr,A_2),hAPP_real_real(log(A_2),X)) = X ) ) ) ) ).

tff(fact_4484_log__powr,axiom,
    ! [B_1: real,Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( hAPP_real_real(log(B_1),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(log(B_1),X)) ) ) ) ).

tff(fact_4485_powr__realpow2,axiom,
    ! [N_1: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( ( ( X = zero_zero_real )
           => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1) = zero_zero_real ) )
          & ( ( X != zero_zero_real )
           => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N_1) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N_1)) ) ) ) ) ) ).

tff(fact_4486_coprime__pow,axiom,
    ! [C_1: nat,N_1: nat,A_2: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_2),B_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_2),B_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,C_1),N_1) )
       => ? [R_1: nat,S_3: nat] :
            ( ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N_1) )
            & ( B_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_3),N_1) ) ) ) ) ).

tff(fact_4487_prime__power__mult,axiom,
    ! [X: nat,Y: nat,K_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) )
       => ? [I_2: nat,J_1: nat] :
            ( ( X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) )
            & ( Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) ) ) ) ) ).

tff(fact_4488_Euler__Fermat,axiom,
    ! [X: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
     => ( ( hAPP_int_int(legacy_zgcd(X),M_1) = one_one_int )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),phi(M_1))),one_one_int),M_1)) ) ) ).

tff(fact_4489_zgcd__zdvd2,axiom,
    ! [I: int,J_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),J_2)) ).

tff(fact_4490_zgcd__zdvd1,axiom,
    ! [I: int,J_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),I)) ).

tff(fact_4491_zgcd__zmult__distrib2__abs,axiom,
    ! [K_2: int,M_1: int,N_1: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(legacy_zgcd(M_1),N_1)) ).

tff(fact_4492_zgcd__pos,axiom,
    ! [I: int,J_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(I),J_2))) ).

tff(fact_4493_zgcd__self,axiom,
    ! [M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
     => ( hAPP_int_int(legacy_zgcd(M_1),M_1) = M_1 ) ) ).

tff(fact_4494_zgcd__geq__zero,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X),Y))) ).

tff(fact_4495_zgcd__0,axiom,
    ! [M_1: int] : hAPP_int_int(legacy_zgcd(M_1),zero_zero_int) = hAPP_int_int(abs_abs_int,M_1) ).

tff(fact_4496_zgcd__0__left,axiom,
    ! [M_1: int] : hAPP_int_int(legacy_zgcd(zero_zero_int),M_1) = hAPP_int_int(abs_abs_int,M_1) ).

tff(fact_4497_zgcd__zdvd__zgcd__zmult,axiom,
    ! [M_1: int,N_1: int,K_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(M_1),N_1)),hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),N_1))) ).

tff(fact_4498_zgcd__1__power__left__distrib,axiom,
    ! [N_1: nat,A_2: int,B_1: int] :
      ( ( hAPP_int_int(legacy_zgcd(A_2),B_1) = one_one_int )
     => ( hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)),B_1) = one_one_int ) ) ).

tff(fact_4499_zgcd__1__power__distrib,axiom,
    ! [N_1: nat,A_2: int,B_1: int] :
      ( ( hAPP_int_int(legacy_zgcd(A_2),B_1) = one_one_int )
     => ( hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),N_1)) = one_one_int ) ) ).

tff(fact_4500_zgcd__power__distrib,axiom,
    ! [A_2: int,B_1: int,N_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_2),B_1)),N_1) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),N_1)) ).

tff(fact_4501_zgcd__commute,axiom,
    ! [I: int,J_2: int] : hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(J_2),I) ).

tff(fact_4502_zgcd__left__commute,axiom,
    ! [K_2: int,M_1: int,N_1: int] : hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M_1),N_1)) = hAPP_int_int(legacy_zgcd(M_1),hAPP_int_int(legacy_zgcd(K_2),N_1)) ).

tff(fact_4503_zgcd__assoc,axiom,
    ! [K_2: int,M_1: int,N_1: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M_1)),N_1) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M_1),N_1)) ).

tff(fact_4504_zgcd__zminus2,axiom,
    ! [I: int,J_2: int] : hAPP_int_int(legacy_zgcd(I),hAPP_int_int(uminus_uminus_int,J_2)) = hAPP_int_int(legacy_zgcd(I),J_2) ).

tff(fact_4505_zgcd__zminus,axiom,
    ! [I: int,J_2: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(uminus_uminus_int,I)),J_2) = hAPP_int_int(legacy_zgcd(I),J_2) ).

tff(fact_4506_zgcd__zadd__zmult,axiom,
    ! [M_1: int,N_1: int,K_2: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),K_2))),N_1) = hAPP_int_int(legacy_zgcd(M_1),N_1) ).

tff(fact_4507_zgcd__greatest,axiom,
    ! [N_1: int,K_2: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M_1),N_1))) ) ) ).

tff(fact_4508_zgcd__greatest__iff,axiom,
    ! [K_1: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(legacy_zgcd(M),N)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) ) ) ).

tff(fact_4509_zgcd__zgcd__zmult,axiom,
    ! [N_1: int,K_2: int,M_1: int] :
      ( ( hAPP_int_int(legacy_zgcd(K_2),M_1) = one_one_int )
     => ( ( hAPP_int_int(legacy_zgcd(N_1),M_1) = one_one_int )
       => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)),M_1) = one_one_int ) ) ) ).

tff(fact_4510_zgcd__zmult__cancel,axiom,
    ! [M_1: int,K_2: int,N_1: int] :
      ( ( hAPP_int_int(legacy_zgcd(K_2),N_1) = one_one_int )
     => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),N_1) = hAPP_int_int(legacy_zgcd(M_1),N_1) ) ) ).

tff(fact_4511_zgcd__1,axiom,
    ! [M_1: int] : hAPP_int_int(legacy_zgcd(M_1),one_one_int) = one_one_int ).

tff(fact_4512_zgcd__1__left,axiom,
    ! [M_1: int] : hAPP_int_int(legacy_zgcd(one_one_int),M_1) = one_one_int ).

tff(fact_4513_zgcd0,axiom,
    ! [I_1: int,J: int] :
      ( ( hAPP_int_int(legacy_zgcd(I_1),J) = zero_zero_int )
    <=> ( ( I_1 = zero_zero_int )
        & ( J = zero_zero_int ) ) ) ).

tff(fact_4514_zgcd__eq,axiom,
    ! [M_1: int,N_1: int] : hAPP_int_int(legacy_zgcd(M_1),N_1) = hAPP_int_int(legacy_zgcd(N_1),hAPP_int_int(div_mod_int(M_1),N_1)) ).

tff(fact_4515_zgcd__zmult__eq__self2,axiom,
    ! [N_1: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)),K_2) = K_2 ) ) ).

tff(fact_4516_zgcd__zmult__distrib2,axiom,
    ! [M_1: int,N_1: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),hAPP_int_int(legacy_zgcd(M_1),N_1)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) ) ) ).

tff(fact_4517_zgcd__zmult__eq__self,axiom,
    ! [N_1: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) = K_2 ) ) ).

tff(fact_4518_zdvd__iff__zgcd,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N))
      <=> ( hAPP_int_int(legacy_zgcd(N),M) = M ) ) ) ).

tff(fact_4519_zgcd__non__0,axiom,
    ! [M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( hAPP_int_int(legacy_zgcd(M_1),N_1) = hAPP_int_int(legacy_zgcd(N_1),hAPP_int_int(div_mod_int(M_1),N_1)) ) ) ).

tff(fact_4520_div__zgcd__relprime,axiom,
    ! [B_1: int,A_2: int] :
      ( ( ( A_2 != zero_zero_int )
        | ( B_1 != zero_zero_int ) )
     => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(legacy_zgcd(A_2),B_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),hAPP_int_int(legacy_zgcd(A_2),B_1))) = one_one_int ) ) ).

tff(fact_4521_zcong__zgcd__zmult__zmod,axiom,
    ! [N_1: int,A_2: int,B_1: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),N_1))
       => ( ( hAPP_int_int(legacy_zgcd(M_1),N_1) = one_one_int )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1))) ) ) ) ).

tff(fact_4522_zgcd__0__1__iff,axiom,
    ! [M: int] :
      ( ( hAPP_int_int(legacy_zgcd(zero_zero_int),M) = one_one_int )
    <=> ( hAPP_int_int(abs_abs_int,M) = one_one_int ) ) ).

tff(fact_4523_zgcd__zmult__zdvd__zgcd,axiom,
    ! [M_1: int,K_2: int,N_1: int] :
      ( ( hAPP_int_int(legacy_zgcd(K_2),N_1) = one_one_int )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),N_1)),hAPP_int_int(legacy_zgcd(M_1),N_1))) ) ).

tff(fact_4524_zrelprime__zdvd__zmult,axiom,
    ! [M_1: int,N_1: int,K_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(N_1),K_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1)) ) ) ).

tff(fact_4525_zrelprime__dvd__mult,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(I),J_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),K_2)) ) ) ).

tff(fact_4526_zgcd__code,axiom,
    ! [K_1: int,L: int] : hAPP_int_int(legacy_zgcd(K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) ).

tff(fact_4527_zprime__imp__zrelprime,axiom,
    ! [N_1: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N_1))
       => ( hAPP_int_int(legacy_zgcd(N_1),P_3) = one_one_int ) ) ) ).

tff(fact_4528_zgcd1__iff__no__common__primedivisor,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(legacy_zgcd(Aa),Ba) = one_one_int )
    <=> ~ ? [P_4: int] :
            ( hBOOL(hAPP_int_bool(zprime,P_4))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),Aa))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),Ba)) ) ) ).

tff(fact_4529_zgcd__zcong__zgcd,axiom,
    ! [B_1: int,A_2: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
     => ( ( hAPP_int_int(legacy_zgcd(A_2),M_1) = one_one_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_2),B_1),M_1))
         => ( hAPP_int_int(legacy_zgcd(B_1),M_1) = one_one_int ) ) ) ) ).

tff(fact_4530_zless__zprime__imp__zrelprime,axiom,
    ! [N_1: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),P_3))
         => ( hAPP_int_int(legacy_zgcd(N_1),P_3) = one_one_int ) ) ) ) ).

tff(fact_4531_zcong__cancel2,axiom,
    ! [Aa: int,Ba: int,K_1: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(K_1),M) = one_one_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Aa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ba)),M))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M)) ) ) ) ).

tff(fact_4532_zcong__cancel,axiom,
    ! [Aa: int,Ba: int,K_1: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(K_1),M) = one_one_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),K_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),K_1)),M))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M)) ) ) ) ).

tff(fact_4533_zrelprime__zdvd__zmult__aux,axiom,
    ! [M_1: int,N_1: int,K_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(N_1),K_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1)) ) ) ) ).

tff(fact_4534_Bnor__mem__if,axiom,
    ! [Aa: int,Ba: int,M: int] :
      ( ( hAPP_int_int(legacy_zgcd(Ba),M) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_i1948725293t_bool(bnorRset(Aa),M))) ) ) ) ).

tff(fact_4535_int__relprime__odd__power__divisors,axiom,
    ! [Aa: int,Ba: int,C: int,X_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat_1,X_1)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat_1,X_1)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
         => ( ( hAPP_int_int(legacy_zgcd(Aa),Ba) = one_one_int )
           => ? [K: int] : Aa = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat_1,X_1)) ) ) ) ) ).

tff(fact_4536_xzgcd__correct,axiom,
    ! [M: int,K_1: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( ( hAPP_int_int(legacy_zgcd(M),N) = K_1 )
      <=> ? [S_3: int,T_1: int] : xzgcd(M,N) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_1)) ) ) ).

tff(fact_4537_int__relprime__power__divisors,axiom,
    ! [A_2: int,B_1: int,C_1: int,N_1: nat] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C_1),N_1) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
       => ( ( hAPP_int_int(legacy_zgcd(A_2),B_1) = one_one_int )
         => ? [K: int] : hAPP_int_int(abs_abs_int,A_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N_1) ) ) ) ).

tff(fact_4538_xzgcd__correct__aux2,axiom,
    ! [M_1: int,N_1: int,R_5: int,R_3: int,S_4: int,S_2: int,T_3: int,T_2: int,K_2: int] :
      ( ? [Sn: int,Tn: int] : xzgcda(M_1,N_1,R_5,R_3,S_4,S_2,T_3,T_2) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
       => ( hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2 ) ) ) ).

tff(fact_4539_zcong__lineq__unique,axiom,
    ! [B_1: int,A_2: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( ( hAPP_int_int(legacy_zgcd(A_2),N_1) = one_one_int )
       => ? [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),N_1))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),X_2)),B_1),N_1))
            & ! [Y_1: int] :
                ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
                  & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),N_1))
                  & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),Y_1)),B_1),N_1)) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_4540_zcong__lineq__ex,axiom,
    ! [A_2: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( ( hAPP_int_int(legacy_zgcd(A_2),N_1) = one_one_int )
       => ? [X_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),X_2)),one_one_int),N_1)) ) ) ).

tff(fact_4541_xzgcd__correct__aux1,axiom,
    ! [M_1: int,N_1: int,S_4: int,S_2: int,T_3: int,T_2: int,R_5: int,R_3: int,K_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
       => ? [Sn: int,Tn: int] : xzgcda(M_1,N_1,R_5,R_3,S_4,S_2,T_3,T_2) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) ) ) ).

tff(fact_4542_int__triple__relprime__odd__power__divisors,axiom,
    ! [Aa: int,Ba: int,C: int,D: int,X_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba)),C) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D),hAPP_int_nat(nat_1,X_1)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat_1,X_1)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
         => ( ( hAPP_int_int(legacy_zgcd(Aa),Ba) = one_one_int )
           => ( ( hAPP_int_int(legacy_zgcd(Ba),C) = one_one_int )
             => ( ( hAPP_int_int(legacy_zgcd(C),Aa) = one_one_int )
               => ? [K: int,L_1: int,M_2: int] :
                    ( ( Aa = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat_1,X_1)) )
                    & ( Ba = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat_1,X_1)) )
                    & ( C = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_2),hAPP_int_nat(nat_1,X_1)) ) ) ) ) ) ) ) ) ).

tff(fact_4543_zgcd__ex__linear,axiom,
    ! [M_1: int,K_2: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( ( hAPP_int_int(legacy_zgcd(M_1),N_1) = K_2 )
       => ? [S_3: int,T_1: int] : K_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N_1)) ) ) ).

tff(fact_4544_int__triple__relprime__power__divisors,axiom,
    ! [A_2: int,B_1: int,C_1: int,D_3: int,N_1: nat] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)),C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_3),N_1) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_1))
       => ( ( hAPP_int_int(legacy_zgcd(A_2),B_1) = one_one_int )
         => ( ( hAPP_int_int(legacy_zgcd(B_1),C_1) = one_one_int )
           => ( ( hAPP_int_int(legacy_zgcd(C_1),A_2) = one_one_int )
             => ? [K: int,L_1: int,M_2: int] :
                  ( ( hAPP_int_int(abs_abs_int,A_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N_1) )
                  & ( hAPP_int_int(abs_abs_int,B_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N_1) )
                  & ( hAPP_int_int(abs_abs_int,C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_2),N_1) ) ) ) ) ) ) ) ).

tff(fact_4545_make__zrelprime,axiom,
    ! [B_1: int,A_2: int] :
      ( ( ( A_2 != zero_zero_int )
        | ( B_1 != zero_zero_int ) )
     => ? [C_3: int,D_2: int] :
          ( ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_2),B_1)),C_3) )
          & ( B_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_2),B_1)),D_2) )
          & ( hAPP_int_int(legacy_zgcd(C_3),D_2) = one_one_int ) ) ) ).

tff(fact_4546_polar__Ex,axiom,
    ! [Y: real,X: real] :
    ? [R_1: real,A_1: real] :
      ( ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_1)) )
      & ( Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_1)) ) ) ).

tff(fact_4547_dist__real__def,axiom,
    ! [X: real,Y: real] : hAPP_real_real(dist_dist_real(X),Y) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) ).

tff(fact_4548_dist__complex__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(dist_dist_complex(X),Y) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) ).

tff(fact_4549_int__of__nat__induct,axiom,
    ! [Z_2: int,P_1: fun_int_bool] :
      ( ! [N_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_2)))
     => ( ! [N_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2)))))
       => hBOOL(hAPP_int_bool(P_1,Z_2)) ) ) ).

tff(fact_4550_int__cases,axiom,
    ! [Z_1: int] :
      ( ! [N_2: nat] : Z_1 != hAPP_nat_int(semiri1621563631at_int,N_2)
     => ~ ! [N_2: nat] : Z_1 != hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_4551_DERIV__nonneg__imp__nonincreasing,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Aa)),hAPP_real_real(F,Ba))) ) ) ).

tff(fact_4552_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),zero_zero_real)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Ba)),hAPP_real_real(F,Aa))) ) ) ).

tff(fact_4553_gcd__nat__induct,axiom,
    ! [M: nat,N: nat,P_1: fun_nat_fun_nat_bool] :
      ( ! [M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_2),zero_zero_nat))
     => ( ! [M_2: nat,N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
           => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),hAPP_nat_nat(div_mod_nat(M_2),N_2)))
             => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_2),N_2)) ) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M),N)) ) ) ).

tff(fact_4554_divides__fact,axiom,
    ! [N_1: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_3),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N_1))) ) ) ).

tff(fact_4555_mod__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,M_1: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(N_1),M_1) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N_1)),quickcheck_int_of(M_1))) ).

tff(fact_4556_Quickcheck__Narrowing_Oof__int__inject,axiom,
    ! [N: int,M: int] :
      ( ( hAPP_i1732201573de_int(quickcheck_of_int,N) = hAPP_i1732201573de_int(quickcheck_of_int,M) )
    <=> ( N = M ) ) ).

tff(fact_4557_int__of__inverse,axiom,
    ! [X: quickcheck_code_int] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X)) = X ).

tff(fact_4558_of__int__int__of,axiom,
    ! [K_2: quickcheck_code_int] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(K_2)) = K_2 ).

tff(fact_4559_int__of__of__int,axiom,
    ! [N_1: int] : quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N_1)) = N_1 ).

tff(fact_4560_number__of__code__int__def,axiom,
    number1226105091de_int = quickcheck_of_int ).

tff(fact_4561_fact__mono,axiom,
    ! [M_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact,M_1)),hAPP_nat_nat(fact,N_1))) ) ).

tff(fact_4562_times__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,M_1: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N_1),M_1) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,quickcheck_int_of(N_1)),quickcheck_int_of(M_1))) ).

tff(fact_4563_plus__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,M_1: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N_1),M_1) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N_1)),quickcheck_int_of(M_1))) ).

tff(fact_4564_minus__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,M_1: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N_1),M_1) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N_1)),quickcheck_int_of(M_1))) ).

tff(fact_4565_zero__code__int__def,axiom,
    zero_z891286103de_int = hAPP_i1732201573de_int(quickcheck_of_int,zero_zero_int) ).

tff(fact_4566_fact__lt,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N_1))) ).

tff(fact_4567_fact_Osimps_I1_J,axiom,
    hAPP_nat_nat(fact,zero_zero_nat) = one_one_nat ).

tff(fact_4568_fact_Osimps_I2_J,axiom,
    ! [N_1: nat] : hAPP_nat_nat(fact,hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(fact,N_1)) ).

tff(fact_4569_fact__le,axiom,
    ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N_1))) ).

tff(fact_4570_one__code__int__def,axiom,
    one_on1684967323de_int = hAPP_i1732201573de_int(quickcheck_of_int,one_one_int) ).

tff(fact_4571_div__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,M_1: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N_1),M_1) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,quickcheck_int_of(N_1)),quickcheck_int_of(M_1))) ).

tff(fact_4572_euclid__bound,axiom,
    ! [N_1: nat] :
    ? [P_4: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_4))
      & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),P_4))
      & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N_1)))) ) ).

tff(fact_4573_weak__decr__stable,axiom,
    ! [F: fun_nat_nat] :
      ( ! [I_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_2))),hAPP_nat_nat(F,I_2)))
     => ? [I_2: nat] :
        ! [K: nat] : hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)) = hAPP_nat_nat(F,I_2) ) ).

tff(fact_4574_norm__frac_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [A_1: int,B_3: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_3)))
           => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_3),zero_zero_int))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(uminus_uminus_int,A_1)),hAPP_int_int(uminus_uminus_int,B_3))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_1),B_3)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4575_upto_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [I_2: int,J_1: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,I_2),J_1)))
           => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)),J_1)) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4576_small__lazy_H_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [D_2: int,I_2: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,D_2),I_2)))
           => ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,D_2),I_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),I_2)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4577_norRRset__def,axiom,
    ! [M: int] : hAPP_i1948725293t_bool(norRRset,M) = hAPP_i1948725293t_bool(bnorRset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),one_one_int)),M) ).

tff(fact_4578_of__int__int__eq,axiom,
    ! [N_1: nat] : hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).

tff(fact_4579_real__eq__of__int,axiom,
    real_int = ring_1_of_int_real ).

tff(fact_4580_real__of__int__def,axiom,
    real_int = ring_1_of_int_real ).

tff(fact_4581_complex__cnj__of__int,axiom,
    ! [Z_1: int] : hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_complex(ring_11397209091omplex,Z_1) ).

tff(fact_4582_complex__Re__of__int,axiom,
    ! [Z_1: int] : hAPP_complex_real(re,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_real(ring_1_of_int_real,Z_1) ).

tff(fact_4583_complex__number__of__def,axiom,
    ! [W: int] : hAPP_int_complex(number528085621omplex,W) = hAPP_int_complex(ring_11397209091omplex,W) ).

tff(fact_4584_int__number__of__def,axiom,
    ! [W: int] : hAPP_int_int(number_number_of_int,W) = hAPP_int_int(ring_1_of_int_int,W) ).

tff(fact_4585_number__of__real__def,axiom,
    ! [X: int] : hAPP_int_real(number267125858f_real,X) = hAPP_int_real(ring_1_of_int_real,X) ).

tff(fact_4586_complex__Im__of__int,axiom,
    ! [Z_1: int] : hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1)) = zero_zero_real ).

tff(fact_4587_phi__def,axiom,
    ! [M: int] : phi(M) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,M)) ).

tff(fact_4588_norR__mem__unique,axiom,
    ! [Aa: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(Aa),M) = one_one_int )
       => ? [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),X_2),M))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(norRRset,M)))
            & ! [Y_1: int] :
                ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Y_1),M))
                  & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_1),hAPP_i1948725293t_bool(norRRset,M))) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_4589_card__nor__eq__noX,axiom,
    ! [X_1: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(X_1),M) = one_one_int )
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(M),X_1)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,M)) ) ) ) ).

tff(fact_4590_noX__is__RRset,axiom,
    ! [X_1: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(X_1),M) = one_one_int )
       => hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(M),X_1)),M)) ) ) ).

tff(fact_4591_RRset__gcd,axiom,
    ! [Aa: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(is_RRset(A),M))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A))
       => ( hAPP_int_int(legacy_zgcd(Aa),M) = one_one_int ) ) ) ).

tff(fact_4592_RRset__zcong__eq,axiom,
    ! [Aa: int,Ba: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(is_RRset(A),M))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),Ba),M))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A))
           => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),A))
             => ( Aa = Ba ) ) ) ) ) ) ).

tff(fact_4593_RRset2norRR__correct,axiom,
    ! [Aa: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(is_RRset(A),M))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),hAPP_int_int(rRset2norRR(A,M),Aa)),M))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,M),Aa)),hAPP_i1948725293t_bool(norRRset,M))) ) ) ) ) ).

tff(fact_4594_RRset2norRR__correct2,axiom,
    ! [Aa: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(is_RRset(A),M))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,M),Aa)),hAPP_i1948725293t_bool(norRRset,M))) ) ) ) ).

tff(fact_4595_RRset2norRR__correct1,axiom,
    ! [Aa: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(is_RRset(A),M))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),hAPP_int_int(rRset2norRR(A,M),Aa)),M)) ) ) ) ).

tff(fact_4596_is__RRset__def,axiom,
    ! [A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(is_RRset(A),M))
    <=> ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(M)))
        & ( hAPP_f957591787ol_nat(finite_card_int,A) = phi(M) ) ) ) ).

tff(fact_4597_RsetR__fin,axiom,
    ! [A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(M)))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ).

tff(fact_4598_Bnor__in__RsetR,axiom,
    ! [Aa: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),M))
     => hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(Aa),M)),rsetR(M))) ) ).

tff(fact_4599_RRset2norRR__eq__norR,axiom,
    ! [A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(is_RRset(A),M))
       => ( image_int_int(rRset2norRR(A,M),A) = hAPP_i1948725293t_bool(norRRset,M) ) ) ) ).

tff(fact_4600_Nitpick_Oint__lcm__def,axiom,
    ! [X: int,Y: int] : hAPP_int_int(int_lcm(X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Y)))) ).

tff(fact_4601_ResSet__image,axiom,
    ! [F: fun_int_int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_f448129468l_bool(resSet(M),A))
       => ( ! [X_2: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
             => ! [Xa: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A))
                 => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X_2)),hAPP_int_int(F,Xa)),M))
                   => ( X_2 = Xa ) ) ) )
         => hBOOL(hAPP_f448129468l_bool(resSet(M),image_int_int(F,A))) ) ) ) ).

tff(fact_4602_transfer__nat__int__set__relations_I1_J,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) ) ).

tff(fact_4603_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [A: fun_nat_bool] : hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A)) ).

tff(fact_4604_transfer__nat__int__set__relations_I2_J,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A))) ) ).

tff(fact_4605_transfer__nat__int__set__relations_I3_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
    <=> ( image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) ) ) ).

tff(fact_4606_transfer__int__nat__set__return__embed,axiom,
    ! [A: fun_nat_bool] : image_int_nat(nat_1,image_nat_int(semiri1621563631at_int,A)) = A ).

tff(fact_4607_transfer__nat__int__set__relations_I4_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

tff(fact_4608_transfer__nat__int__set__relations_I5_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

tff(fact_4609_SetS__def,axiom,
    ! [Aa: int,Pa: int] : setS(Aa,Pa) = image_275383677t_bool(multInvPair(Aa,Pa),hAPP_i1948725293t_bool(sRStar,Pa)) ).

tff(fact_4610_infinite__nat__iff__unbounded__le,axiom,
    ! [S: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ! [M_2: nat] :
        ? [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
          & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) ) ) ).

tff(fact_4611_infinite__nat__iff__unbounded,axiom,
    ! [S: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ! [M_2: nat] :
        ? [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
          & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) ) ) ).

tff(fact_4612_Nitpick_Onat__lcm__def,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(nat_lcm(X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(nat_gcd(X),Y)) ).

tff(fact_4613_transfer__int__nat__set__relations_I4_J,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A),B))
        <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat_1,A)),image_int_nat(nat_1,B))) ) ) ) ).

tff(fact_4614_gcd__eq__nitpick__gcd,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(nat_gcd(X),Y) ).

tff(fact_4615_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ) ).

tff(fact_4616_nat__set__def,axiom,
    ! [S: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,S))
    <=> ! [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),S))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) ) ) ).

tff(fact_4617_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,
    ! [C_2: fun_nat_bool] : hBOOL(hAPP_f448129468l_bool(nat_nat_set,image_nat_int(semiri1621563631at_int,C_2))) ).

tff(fact_4618_transfer__int__nat__set__relations_I3_J,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B))
       => ( ( A = B )
        <=> ( image_int_nat(nat_1,A) = image_int_nat(nat_1,B) ) ) ) ) ).

tff(fact_4619_nat__gcd_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( hAPP_nat_nat(nat_gcd(X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( hAPP_nat_nat(nat_gcd(X),Y) = hAPP_nat_nat(nat_gcd(Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ) ).

tff(fact_4620_transfer__nat__int__set__return__embed,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( image_nat_int(semiri1621563631at_int,image_int_nat(nat_1,A)) = A ) ) ).

tff(fact_4621_transfer__int__nat__set__relations_I1_J,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
      <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat_1,A))) ) ) ).

tff(fact_4622_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hAPP_f957591787ol_nat(finite_card_int,A) = hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat_1,A)) ) ) ).

tff(fact_4623_transfer__int__nat__set__relations_I5_J,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
        <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat_1,A)),image_int_nat(nat_1,B))) ) ) ) ).

tff(fact_4624_nat__gcd_Opsimps,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_1),Y_2)))
     => ( ( ( Y_2 = zero_zero_nat )
         => ( hAPP_nat_nat(nat_gcd(X_1),Y_2) = X_1 ) )
        & ( ( Y_2 != zero_zero_nat )
         => ( hAPP_nat_nat(nat_gcd(X_1),Y_2) = hAPP_nat_nat(nat_gcd(Y_2),hAPP_nat_nat(div_mod_nat(X_1),Y_2)) ) ) ) ) ).

tff(fact_4625_Nitpick_Oint__gcd__def,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(int_gcd,X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_gcd(hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Y)))) ).

tff(fact_4626_nat__gcd_Opinduct,axiom,
    ! [P_1: fun_nat_fun_nat_bool,A0: nat,A1: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A0),A1)))
     => ( ! [X_2: nat,Y_1: nat] :
            ( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_2),Y_1)))
           => ( ( ( Y_1 != zero_zero_nat )
               => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y_1),hAPP_nat_nat(div_mod_nat(X_2),Y_1))) )
             => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X_2),Y_1)) ) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4627_finite__nat__set__iff__bounded__le,axiom,
    ! [N_4: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4))
    <=> ? [M_2: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_2)) ) ) ).

tff(fact_4628_finite__nat__set__iff__bounded,axiom,
    ! [N_4: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4))
    <=> ? [M_2: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_2)) ) ) ).

tff(fact_4629_bounded__nat__set__is__finite,axiom,
    ! [N: nat,N_4: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),N)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)) ) ).

tff(fact_4630_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

tff(fact_4631_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U))) ).

tff(fact_4632_Rats__abs__nat__div__natE,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real))
     => ~ ! [M_2: nat,N_2: nat] :
            ( ( N_2 != zero_zero_nat )
           => ( ( hAPP_real_real(abs_abs_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_2)),hAPP_nat_real(real_nat,N_2)) )
             => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_2),N_2) != one_one_nat ) ) ) ) ).

tff(fact_4633_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U))) ).

tff(fact_4634_Rats__real__nat,axiom,
    ! [N: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N)),field_1210416355s_real)) ).

tff(fact_4635_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),hAPP_nat_nat(suc,L)) ).

tff(fact_4636_Rats__dense__in__nn__real,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
       => ? [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),X_2))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ) ).

tff(fact_4637_DERIV__isconst3,axiom,
    ! [F: fun_real_real,Y_2: real,X_1: real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(Aa),Ba)))
       => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Y_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(Aa),Ba)))
         => ( ! [X_2: real] :
                ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(Aa),Ba)))
               => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
           => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_2) ) ) ) ) ) ).

tff(fact_4638_iszero__rat,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K_1)))
    <=> hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K_1))) ) ).

tff(fact_4639_Rats__dense__in__real,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
     => ? [X_2: real] :
          ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ).

tff(fact_4640_Sup__greaterThanLessThan,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
     => ( comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Y_2),X_1)) = X_1 ) ) ).

tff(fact_4641_Sup__upper__EX,axiom,
    ! [X_1: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3))
     => ( ? [Z: real] :
          ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z)) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),comple124823625p_real(X_3))) ) ) ).

tff(fact_4642_SupInf_OSup__upper,axiom,
    ! [Z_2: real,X_1: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3))
     => ( ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),comple124823625p_real(X_3))) ) ) ).

tff(fact_4643_Frct__code__post_I4_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int)) = hAPP_int_rat(number_number_of_rat,K_2) ).

tff(fact_4644_Frct__code__post_I2_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,K_2),zero_zero_int)) = zero_zero_rat ).

tff(fact_4645_Frct__code__post_I1_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_2)) = zero_zero_rat ).

tff(fact_4646_Frct__code__post_I3_J,axiom,
    frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

tff(fact_4647_Frct__code__post_I6_J,axiom,
    ! [K_2: int,L_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) ).

tff(fact_4648_Frct__code__post_I5_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),hAPP_int_int(number_number_of_int,K_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) ).

tff(fact_4649_Sup__eq__maximum,axiom,
    ! [Z_2: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_3))
     => ( ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)) )
       => ( comple124823625p_real(X_3) = Z_2 ) ) ) ).

tff(fact_4650_SupInf_OSup__upper2,axiom,
    ! [Z_2: real,Y_2: real,X_1: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1))
       => ( ! [X_2: real] :
              ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)) )
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),comple124823625p_real(X_3))) ) ) ) ).

tff(fact_4651_abs__rat__def,axiom,
    ! [Q_1: rat] :
      ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_1),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,Q_1) = hAPP_rat_rat(uminus_uminus_rat,Q_1) ) )
      & ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_1),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,Q_1) = Q_1 ) ) ) ).

tff(fact_4652_sgn__rat__def,axiom,
    ! [Q_1: rat] :
      ( ( ( Q_1 = zero_zero_rat )
       => ( hAPP_rat_rat(sgn_sgn_rat,Q_1) = zero_zero_rat ) )
      & ( ( Q_1 != zero_zero_rat )
       => ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_1))
           => ( hAPP_rat_rat(sgn_sgn_rat,Q_1) = one_one_rat ) )
          & ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_1))
           => ( hAPP_rat_rat(sgn_sgn_rat,Q_1) = hAPP_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

tff(fact_4653_divide__rat__def,axiom,
    ! [Q_1: rat,R_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_1),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_1),hAPP_rat_rat(inverse_inverse_rat,R_3)) ).

tff(fact_4654_normalize__negative,axiom,
    ! [P_3: int,Q_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_1),zero_zero_int))
     => ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(uminus_uminus_int,P_3)),hAPP_int_int(uminus_uminus_int,Q_1))) ) ) ).

tff(fact_4655_rat__sgn__code,axiom,
    ! [P_3: rat] : quotient_of(hAPP_rat_rat(sgn_sgn_rat,P_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(sgn_sgn_int,hAPP_P1175774780nt_int(product_fst_int_int,quotient_of(P_3)))),one_one_int) ).

tff(fact_4656_diff__rat__def,axiom,
    ! [Q_1: rat,R_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_1),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_1),hAPP_rat_rat(uminus_uminus_rat,R_3)) ).

tff(fact_4657_quotient__of__inject__eq,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( quotient_of(Aa) = quotient_of(Ba) )
    <=> ( Aa = Ba ) ) ).

tff(fact_4658_quotient__of__inject,axiom,
    ! [A_2: rat,B_1: rat] :
      ( ( quotient_of(A_2) = quotient_of(B_1) )
     => ( A_2 = B_1 ) ) ).

tff(fact_4659_less__rat__def,axiom,
    ! [Z_2: rat,W_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),W_1))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),W_1))
        & ( Z_2 != W_1 ) ) ) ).

tff(fact_4660_quotient__of__denom__pos,axiom,
    ! [R_3: rat,P_3: int,Q_1: int] :
      ( ( quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)) ) ).

tff(fact_4661_rat__one__code,axiom,
    quotient_of(one_one_rat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

tff(fact_4662_normalize__denom__pos,axiom,
    ! [R_3: product_prod_int_int,P_3: int,Q_1: int] :
      ( ( normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)) ) ).

tff(fact_4663_normalize__denom__zero,axiom,
    ! [P_3: int] : normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),zero_zero_int)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

tff(fact_4664_normalize__crossproduct,axiom,
    ! [P_3: int,R_3: int,S_2: int,Q_1: int] :
      ( ( Q_1 != zero_zero_int )
     => ( ( S_2 != zero_zero_int )
       => ( ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_3),S_2)) )
         => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,P_3),S_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),Q_1) ) ) ) ) ).

tff(fact_4665_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

tff(fact_4666_quotient__of__number_I3_J,axiom,
    ! [K_2: int] : quotient_of(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int) ).

tff(fact_4667_obtain__pos__sum,axiom,
    ! [R_3: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_3))
     => ~ ! [S_3: rat] :
            ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),S_3))
           => ! [T_1: rat] :
                ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),T_1))
               => ( R_3 != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_3),T_1) ) ) ) ) ).

tff(fact_4668_ratrel__iff,axiom,
    ! [X_1: product_prod_int_int,Y_2: product_prod_int_int] :
      ( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_1),Y_2)),ratrel))
    <=> ( ( hAPP_P1175774780nt_int(product_snd_int_int,X_1) != zero_zero_int )
        & ( hAPP_P1175774780nt_int(product_snd_int_int,Y_2) != zero_zero_int )
        & ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_1)) ) ) ) ).

tff(fact_4669_Fract__1__number__of,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),hAPP_int_int(number_number_of_int,K_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) ).

tff(fact_4670_Ratreal__number__of__quotient2,axiom,
    ! [R_3: int,S_2: int] : ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_3)),hAPP_int_rat(number_number_of_rat,S_2))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_2)) ).

tff(fact_4671_quotient__of__eq,axiom,
    ! [A_2: int,B_1: int,P_3: int,Q_1: int] :
      ( ( quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1) ) ) ).

tff(fact_4672_minus__rat,axiom,
    ! [A_2: int,B_1: int] : hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_2)),B_1) ).

tff(fact_4673_divide__rat,axiom,
    ! [A_2: int,B_1: int,C_1: int,D_3: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C_1)) ).

tff(fact_4674_Fract__of__int__quotient,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),L_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(ring_1_of_int_rat,K_2)),hAPP_int_rat(ring_1_of_int_rat,L_2)) ).

tff(fact_4675_Fract__of__int__eq,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) = hAPP_int_rat(ring_1_of_int_rat,K_2) ).

tff(fact_4676_of__int__rat,axiom,
    ! [K_2: int] : hAPP_int_rat(ring_1_of_int_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) ).

tff(fact_4677_rat__number__collapse_I1_J,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),K_2) = zero_zero_rat ).

tff(fact_4678_eq__rat_I3_J,axiom,
    ! [A_2: int,C_1: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),A_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),C_1) ).

tff(fact_4679_rat__number__collapse_I4_J,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),zero_zero_int) = zero_zero_rat ).

tff(fact_4680_one__rat,axiom,
    one_one_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) ).

tff(fact_4681_rat__number__collapse_I2_J,axiom,
    hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) = one_one_rat ).

tff(fact_4682_eq__rat_I2_J,axiom,
    ! [A_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),zero_zero_int) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

tff(fact_4683_Fract__of__nat__eq,axiom,
    ! [K_2: nat] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) = hAPP_nat_rat(semiri151668891at_rat,K_2) ).

tff(fact_4684_of__nat__rat,axiom,
    ! [K_2: nat] : hAPP_nat_rat(semiri151668891at_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) ).

tff(fact_4685_rat__number__of__def,axiom,
    ! [W: int] : hAPP_int_rat(number_number_of_rat,W) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,W),one_one_int) ).

tff(fact_4686_real__floor__code,axiom,
    ! [X: rat] : archim1246769320r_real(ratreal(X)) = archim791455193or_rat(X) ).

tff(fact_4687_floor__Fract,axiom,
    ! [A_2: int,B_1: int] : archim791455193or_rat(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1) ).

tff(fact_4688_inverse__rat,axiom,
    ! [A_2: int,B_1: int] : hAPP_rat_rat(inverse_inverse_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,B_1),A_2) ).

tff(fact_4689_minus__rat__cancel,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_1)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1) ).

tff(fact_4690_mult__rat__cancel,axiom,
    ! [A_2: int,B_1: int,C_1: int] :
      ( ( C_1 != zero_zero_int )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),A_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_1)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1) ) ) ).

tff(fact_4691_eq__rat_I1_J,axiom,
    ! [Aa: int,C: int,D: int,Ba: int] :
      ( ( Ba != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D) )
        <=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),D) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba) ) ) ) ) ).

tff(fact_4692_mult__rat,axiom,
    ! [A_2: int,B_1: int,C_1: int,D_3: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_3)) ).

tff(fact_4693_abs__rat,axiom,
    ! [A_2: int,B_1: int] : hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(abs_abs_int,A_2)),hAPP_int_int(abs_abs_int,B_1)) ).

tff(fact_4694_normalize__eq,axiom,
    ! [A_2: int,B_1: int,P_3: int,Q_1: int] :
      ( ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_1)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1) ) ) ).

tff(fact_4695_real__less__code,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(X_1)),ratreal(Y_2)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2)) ) ).

tff(fact_4696_zero__real__code,axiom,
    zero_zero_real = ratreal(zero_zero_rat) ).

tff(fact_4697_Ratreal__number__collapse_I1_J,axiom,
    ratreal(zero_zero_rat) = zero_zero_real ).

tff(fact_4698_number__of__real__code,axiom,
    ! [K_2: int] : hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2)) ).

tff(fact_4699_Ratreal__number__collapse_I3_J,axiom,
    ! [K_2: int] : ratreal(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_int_real(number267125858f_real,K_2) ).

tff(fact_4700_one__real__code,axiom,
    one_one_real = ratreal(one_one_rat) ).

tff(fact_4701_Ratreal__number__collapse_I2_J,axiom,
    ratreal(one_one_rat) = one_one_real ).

tff(fact_4702_real__less__eq__code,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(X_1)),ratreal(Y_2)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2)) ) ).

tff(fact_4703_real__plus__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Y)) ).

tff(fact_4704_real__minus__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),Y)) ).

tff(fact_4705_real__uminus__code,axiom,
    ! [X: rat] : hAPP_real_real(uminus_uminus_real,ratreal(X)) = ratreal(hAPP_rat_rat(uminus_uminus_rat,X)) ).

tff(fact_4706_real__divide__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X),Y)) ).

tff(fact_4707_real__times__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X),Y)) ).

tff(fact_4708_real__inverse__code,axiom,
    ! [X: rat] : hAPP_real_real(inverse_inverse_real,ratreal(X)) = ratreal(hAPP_rat_rat(inverse_inverse_rat,X)) ).

tff(fact_4709_zero__rat,axiom,
    zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

tff(fact_4710_rat__number__expand_I3_J,axiom,
    ! [K_2: int] : hAPP_int_rat(number_number_of_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) ).

tff(fact_4711_rat__number__collapse_I3_J,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = hAPP_int_rat(number_number_of_rat,K_2) ).

tff(fact_4712_Fract__number__of__quotient,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) ).

tff(fact_4713_quotient__of__Fract,axiom,
    ! [A_2: int,B_1: int] : quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_1)) ).

tff(fact_4714_Frct__def,axiom,
    ! [P_3: product_prod_int_int] : frct(P_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)) ).

tff(fact_4715_less__rat,axiom,
    ! [Aa: int,C: int,D: int,Ba: int] :
      ( ( Ba != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D)))) ) ) ) ).

tff(fact_4716_Ratreal__number__of__quotient,axiom,
    ! [R_3: int,S_2: int] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_3))),ratreal(hAPP_int_rat(number_number_of_rat,S_2))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_2)) ).

tff(fact_4717_sgn__rat,axiom,
    ! [A_2: int,B_1: int] : hAPP_rat_rat(sgn_sgn_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)) = hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,A_2)),hAPP_int_int(sgn_sgn_int,B_1))) ).

tff(fact_4718_Fract__less__zero__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)),zero_zero_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ) ).

tff(fact_4719_zero__less__Fract__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ) ).

tff(fact_4720_le__rat,axiom,
    ! [Aa: int,C: int,D: int,Ba: int] :
      ( ( Ba != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),Ba)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D)))) ) ) ) ).

tff(fact_4721_add__rat,axiom,
    ! [A_2: int,C_1: int,D_3: int,B_1: int] :
      ( ( B_1 != zero_zero_int )
     => ( ( D_3 != zero_zero_int )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_3)) ) ) ) ).

tff(fact_4722_diff__rat,axiom,
    ! [A_2: int,C_1: int,D_3: int,B_1: int] :
      ( ( B_1 != zero_zero_int )
     => ( ( D_3 != zero_zero_int )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_3)) ) ) ) ).

tff(fact_4723_one__less__Fract__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ).

tff(fact_4724_Fract__less__one__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)),one_one_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ) ).

tff(fact_4725_Fract__add__one,axiom,
    ! [M_1: int,N_1: int] :
      ( ( N_1 != zero_zero_int )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_1),N_1)),N_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,M_1),N_1)),one_one_rat) ) ) ).

tff(fact_4726_Fract__le__zero__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)),zero_zero_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ) ).

tff(fact_4727_zero__le__Fract__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ) ).

tff(fact_4728_rat__floor__lemma,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1))),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)))
      & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1)),hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),B_1)),one_one_int)))) ) ).

tff(fact_4729_one__le__Fract__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa)) ) ) ).

tff(fact_4730_Fract__le__one__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,Aa),Ba)),one_one_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ) ).

tff(fact_4731_Rat__induct__pos,axiom,
    ! [Qa: rat,P_1: fun_rat_bool] :
      ( ! [A_1: int,B_3: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_3))
         => hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_3))) )
     => hBOOL(hAPP_rat_bool(P_1,Qa)) ) ).

tff(fact_4732_adjust__eq,axiom,
    ! [Ba: int,Qa: int,R_2: int] : hAPP_P1975530577nt_int(adjust(Ba),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Qa),R_2)) = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),Ba))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Qa)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),Ba))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Qa)),R_2)) ).

tff(fact_4733_lemma__DERIV__sin__cos__minus,axiom,
    ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),uminus_uminus_real))),sin))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),uminus_uminus_real))),cos))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real)) ).

tff(fact_4734_finite__Collect__less__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1)))) ).

tff(fact_4735_finite__Collect__le__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1)))) ).

tff(fact_4736_transfer__morphism__int__nat,axiom,
    hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(semiri1621563631at_int),cOMBK_bool_nat(fTrue))) ).

tff(fact_4737_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
    ! [P_1: fun_int_bool] : collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = image_nat_int(semiri1621563631at_int,collect_nat(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P_1),semiri1621563631at_int))) ).

tff(fact_4738_int__card__bdd__int__set__l__le,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) = N ) ) ).

tff(fact_4739_card__bdd__int__set__l,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))) = hAPP_int_nat(nat_1,N) ) ) ).

tff(fact_4740_card__bdd__int__set__l__le,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))) = hAPP_int_nat(nat_1,N) ) ) ).

tff(fact_4741_bdd__int__set__l__le__finite,axiom,
    ! [N: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) ).

tff(fact_4742_bdd__int__set__l__finite,axiom,
    ! [N: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) ).

tff(fact_4743_DERIV__cos__add,axiom,
    ! [K_1: real,Xa_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1))))) ).

tff(fact_4744_less__eq__Suc__le__raw,axiom,
    ! [X_2: nat] : hAPP_n1699378549t_bool(ord_less_nat,X_2) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_2)) ).

tff(fact_4745_DERIV__sin__sin__mult,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))))) ).

tff(fact_4746_DERIV__sin__add,axiom,
    ! [K_1: real,Xa_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1)))) ).

tff(fact_4747_card__bdd__nat__set__le,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_1))) = hAPP_nat_nat(suc,X_1) ).

tff(fact_4748_card__Collect__le__nat,axiom,
    ! [N: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),N))) = hAPP_nat_nat(suc,N) ).

tff(fact_4749_bdd__nat__set__le__finite,axiom,
    ! [X_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_1)))) ).

tff(fact_4750_nat__number__of__Bit0,axiom,
    ! [W: int] : hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W)) ).

tff(fact_4751_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J,axiom,
    ! [P_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(nat_nat_set,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)))) ).

tff(fact_4752_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J,axiom,
    ! [P_1: fun_nat_bool] : collect_nat(P_1) = image_int_nat(nat_1,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P_1),nat_1)))) ).

tff(fact_4753_bdd__int__set__le__finite,axiom,
    ! [N: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N))))) ).

tff(fact_4754_Bseq__realpow,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))
       => hBOOL(bseq_real(hAPP_r474017924t_real(power_power_real,X_1))) ) ) ).

tff(fact_4755_diff__nat__eq__if,axiom,
    ! [Z_2: int,Z_3: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,Z_3))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,Z_2)),hAPP_int_nat(nat_1,Z_3)) = hAPP_int_nat(nat_1,Z_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,Z_3))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,Z_2)),hAPP_int_nat(nat_1,Z_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))),zero_zero_nat),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))) ) ) ) ).

tff(fact_4756_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( M != zero_zero_nat )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,dvd_dvd_nat),M)))) ) ).

tff(fact_4757_finite__divisors__int,axiom,
    ! [I_1: int] :
      ( ( I_1 != zero_zero_int )
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,dvd_dvd_int),I_1)))) ) ).

tff(fact_4758_primes__infinite,axiom,
    ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(prime))) ).

tff(fact_4759_cnj_OCauchy,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(cauchy_complex(X_3))
     => hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))) ) ).

tff(fact_4760_Re_OCauchy,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(cauchy_complex(X_3))
     => hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))) ) ).

tff(fact_4761_Im_OCauchy,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(cauchy_complex(X_3))
     => hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))) ) ).

tff(fact_4762_DERIV__fun__cos,axiom,
    ! [G: fun_real_real,X_1: real,M: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(G,X_1),M))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),G),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,X_1)))),M))) ) ).

tff(fact_4763_DERIV__fun__sin,axiom,
    ! [G: fun_real_real,X_1: real,M: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(G,X_1),M))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,X_1))),M))) ) ).

tff(fact_4764_card__Collect__less__nat,axiom,
    ! [N: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N))) = N ).

tff(fact_4765_card__bdd__nat__set__l,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_1))) = X_1 ).

tff(fact_4766_bdd__nat__set__l__finite,axiom,
    ! [X_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_1)))) ).

tff(fact_4767_finite__M__bounded__by__nat,axiom,
    ! [P_1: fun_nat_bool,I_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),P_1)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) ).

tff(fact_4768_zpower__number__of__even,axiom,
    ! [Z_1: int,W: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) ).

tff(fact_4769_card__less__Suc2,axiom,
    ! [I_1: nat,M_3: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_3))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_3))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) ) ) ).

tff(fact_4770_card__less,axiom,
    ! [I_1: nat,M_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_3))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) != zero_zero_nat ) ) ).

tff(fact_4771_card__less__Suc,axiom,
    ! [I_1: nat,M_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => ( hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_3))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_3))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) ) ) ).

tff(fact_4772_bdd__int__set__l__l__finite,axiom,
    ! [N: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) ).

tff(fact_4773_noXRRset__def,axiom,
    ! [M: int,X_1: int] : hAPP_i1948725293t_bool(noXRRset(M),X_1) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),hAPP_i1948725293t_bool(norRRset,M)) ).

tff(fact_4774_DERIV__fun__pow,axiom,
    ! [N: nat,G: fun_real_real,X_1: real,M: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(G,X_1),M))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),G)),N),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M))) ) ).

tff(fact_4775_card__bdd__int__set__le,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,N)),one_one_nat) ) ) ).

tff(fact_4776_int__card__bdd__int__set__l__l,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N))))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int) ) ) ).

tff(fact_4777_card__bdd__int__set__l__l,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,N)),one_one_nat) ) ) ).

tff(fact_4778_DERIV__cos__cos__mult,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))))) ).

tff(fact_4779_eq__number__of__Suc,axiom,
    ! [V: int,N: nat] :
      ( ( hAPP_int_nat(number_number_of_nat,V) = hAPP_nat_nat(suc,N) )
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => ( hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N ) ) ) ) ).

tff(fact_4780_Suc__eq__number__of,axiom,
    ! [N: nat,V: int] :
      ( ( hAPP_nat_nat(suc,N) = hAPP_int_nat(number_number_of_nat,V) )
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => ( hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N ) ) ) ) ).

tff(fact_4781_nat__number__of__Bit1,axiom,
    ! [W: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) ) ) ) ).

tff(fact_4782_DERIV__pow,axiom,
    ! [N: nat,X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))))) ).

tff(fact_4783_DERIV__cos__cos__mult2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4784_less__number__of__Suc,axiom,
    ! [V: int,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N)) ) ) ).

tff(fact_4785_less__Suc__number__of,axiom,
    ! [N: nat,V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))) ) ) ) ).

tff(fact_4786_le__Suc__number__of,axiom,
    ! [N: nat,V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))) ) ) ) ).

tff(fact_4787_le__number__of__Suc,axiom,
    ! [V: int,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N)) ) ) ).

tff(fact_4788_SR__pos,axiom,
    ! [X_3: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(standardRes(M),X_3)),collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),M))))) ) ).

tff(fact_4789_RsetR__zmult__mono,axiom,
    ! [X_1: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(M)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
       => ( ( hAPP_int_int(legacy_zgcd(X_1),M) = one_one_int )
         => hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),A)),rsetR(M))) ) ) ) ).

tff(fact_4790_zpower__number__of__odd,axiom,
    ! [Z_1: int,W: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = one_one_int ) ) ) ).

tff(fact_4791_DERIV__sin__sin__mult2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4792_DERIV__log,axiom,
    ! [Ba: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(log(Ba),X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(ln,Ba)),X_1)))) ) ).

tff(fact_4793_DERIV__cos__cos__mult3,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) ).

tff(fact_4794_DERIV__sin__circle__all__zero,axiom,
    ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real)) ).

tff(fact_4795_DERIV__sin__realpow2a,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4796_DERIV__sin__realpow2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))))) ).

tff(fact_4797_diff__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,V) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))),zero_zero_nat),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))) ) ) ) ).

tff(fact_4798_DERIV__cos__realpow2a,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4799_DERIV__cos__realpow2b,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) ).

tff(fact_4800_DERIV__cos__realpow2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))))) ).

tff(fact_4801_lemma__DERIV__tan,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) != zero_zero_real )
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),sin)),cos),X_1),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_4802_lemma__DERIV__sin__cos__add,axiom,
    ! [Y_2: real,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real)) ).

tff(fact_4803_DERIV__sin__circle__all,axiom,
    ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) ).

tff(fact_4804_norm__frac_Opsimps,axiom,
    ! [Aa: int,Ba: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aa),Ba)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),zero_zero_int))
         => ( norm_frac(Aa,Ba) = norm_frac(hAPP_int_int(uminus_uminus_int,Aa),hAPP_int_int(uminus_uminus_int,Ba)) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),zero_zero_int))
         => ( ( ( ( Aa = zero_zero_int )
                | ( Ba = zero_zero_int ) )
             => ( norm_frac(Aa,Ba) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
            & ( ~ ( ( Aa = zero_zero_int )
                  | ( Ba = zero_zero_int ) )
             => ( norm_frac(Aa,Ba) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,Aa),Ba))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Ba),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,Aa),Ba))) ) ) ) ) ) ) ).

tff(fact_4805_monoseq__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(monoseq_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) ) ).

tff(fact_4806_monoseq__realpow,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))
       => hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,X_1))) ) ) ).

tff(fact_4807_norm__frac_Osimps,axiom,
    ! [A_2: int,B_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
       => ( norm_frac(A_2,B_1) = norm_frac(hAPP_int_int(uminus_uminus_int,A_2),hAPP_int_int(uminus_uminus_int,B_1)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
       => ( ( ( ( A_2 = zero_zero_int )
              | ( B_1 = zero_zero_int ) )
           => ( norm_frac(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
          & ( ~ ( ( A_2 = zero_zero_int )
                | ( B_1 = zero_zero_int ) )
           => ( norm_frac(A_2,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_2),B_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_2),B_1))) ) ) ) ) ) ).

tff(fact_4808_arg__def,axiom,
    ! [Z_2: complex] : arg(Z_2) = hilbert_Eps_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),cos))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),sin))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,pi)))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))))) ).

tff(fact_4809_Union__SetS__setprod__prop1,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Pa),Aa))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(Aa,Pa)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Aa),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Pa)) ) ) ) ) ).

tff(fact_4810_d22set__prod__zfact,axiom,
    ! [Aa: int] : big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,Aa)) = hAPP_int_int(zfact,Aa) ).

tff(fact_4811_Bnor__prod__zgcd,axiom,
    ! [Aa: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),M))
     => ( hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(Aa),M))),M) = one_one_int ) ) ).

tff(fact_4812_Bnor__prod__power,axiom,
    ! [Aa: int,M: int,X_1: int] :
      ( ( X_1 != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),M))
       => ( big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),hAPP_i1948725293t_bool(bnorRset(Aa),M))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(Aa),M))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(Aa),M)))) ) ) ) ).

tff(fact_4813_SetS__setprod__prop,axiom,
    ! [X_1: fun_int_bool,Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Pa),Aa))
           => ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(Aa,Pa)))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,X_1)),Aa),Pa)) ) ) ) ) ) ).

tff(fact_4814_wset__zcong__prod__1,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Pa))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(wset(Aa),Pa))),one_one_int),Pa)) ) ) ) ).

tff(fact_4815_Union__SetS__setprod__prop2,axiom,
    ! [Aa: int,Pa: int] :
      ( hBOOL(hAPP_int_bool(zprime,Pa))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Pa))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),zero_zero_int),Pa))
         => ( big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(Aa,Pa))) = hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Pa),one_one_int)) ) ) ) ) ).

tff(fact_4816_transfer__nat__int__sum__prod__closure_I2_J,axiom,
    ! [F: fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_2))) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A))) ) ) ).

tff(fact_4817_arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => ( hAPP_real_real(arctan,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) ) ) ).

tff(fact_4818_DERIV__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))))),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),X_1),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ).

tff(fact_4819_sin__def,axiom,
    ! [X_1: real] : hAPP_real_real(sin,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) ).

tff(fact_4820_lemma__sin__ext,axiom,
    ! [X_2: real] : hAPP_real_real(sin,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) ).

tff(fact_4821_cos__def,axiom,
    ! [X_1: real] : hAPP_real_real(cos,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) ).

tff(fact_4822_lemma__cos__ext,axiom,
    ! [X_2: real] : hAPP_real_real(cos,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) ).

tff(fact_4823_lemma__sin__minus,axiom,
    ! [X_1: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4824_exp__first__two__terms,axiom,
    ! [X_1: real] : hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_4825_pi__series,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),one_one_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))) ).

tff(fact_4826_ln__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => ( hAPP_real_real(ln,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real))),suc))) ) ) ) ).

tff(fact_4827_summable__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))) ) ).

tff(fact_4828_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))
     => hBOOL(summable_real(F)) ) ).

tff(fact_4829_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N_3: nat] :
        ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2))) )
     => ( hBOOL(summable_real(G))
       => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) ) ) ).

tff(fact_4830_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,F))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))) ) ).

tff(fact_4831_summable__sin,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4832_summable__cos,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4833_rat__floor__code,axiom,
    ! [Pa: rat] : archim791455193or_rat(Pa) = hAPP_P1175774780nt_int(produc1298267108nt_int(div_div_int),quotient_of(Pa)) ).

tff(fact_4834_rat__uminus__code,axiom,
    ! [Pa: rat] : quotient_of(hAPP_rat_rat(uminus_uminus_rat,Pa)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),quotient_of(Pa)) ).

tff(fact_4835_summable__exp,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4836_rat__abs__code,axiom,
    ! [Pa: rat] : quotient_of(hAPP_rat_rat(abs_abs_rat,Pa)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),abs_abs_int)),quotient_of(Pa)) ).

tff(fact_4837_divmod__int__rel__def,axiom,
    ! [Aa: int,Ba: int] : divmod_int_rel(Aa,Ba) = produc93441119t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool),hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj)),hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(fequal_int,Aa))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,Ba)))))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Ba))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_bool_bool(fNot,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ba)))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,Ba))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),zero_zero_int)))))) ).

tff(fact_4838_sum2sq__def,axiom,
    twoSqu2000912462sum2sq = produc1298267108nt_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,cOMBB_int_int_int),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,power_power_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,power_power_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4839_rat__inverse__code,axiom,
    ! [Pa: rat] : quotient_of(hAPP_rat_rat(inverse_inverse_rat,Pa)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,cOMBB_1389251822nt_int),hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)))),hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,cOMBC_1964283556nt_int),hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int)),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,times_times_int),sgn_sgn_int)))),abs_abs_int))),quotient_of(Pa)) ).

tff(fact_4840_exp__tail__after__first__two__terms__summable,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_4841_adjust__def,axiom,
    ! [Ba: int] : adjust(Ba) = produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),Ba))))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),Ba))))),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_4842_DERIV__power__series_H,axiom,
    ! [X0: real,F: fun_nat_real,R_4: real] :
      ( ! [X_2: real] :
          ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4)))
         => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X_2)))) )
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4)))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_4))
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),suc))),X0),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X0))))) ) ) ) ).

tff(fact_4843_Nitpick_OFrac__def,axiom,
    frac = produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)))),hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,cOMBC_int_int_bool),hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int)),int_gcd))),one_one_int))) ).

tff(fact_4844_cnj_Osuminf,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => ( hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)) ) ) ).

tff(fact_4845_cnj_Osummable,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))) ) ).

tff(fact_4846_Re_Osummable,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))) ) ).

tff(fact_4847_Im_Osummable,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))) ) ).

tff(fact_4848_Re_Osuminf,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => ( hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)) ) ) ).

tff(fact_4849_Im_Osuminf,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => ( hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)) ) ) ).

tff(fact_4850_ratrel__def,axiom,
    ratrel = collec50511176nt_int(produc973645403t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,cOMBB_1344650379nt_int),hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,fNot),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),product_snd_int_int)),zero_zero_int))))),hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,fNot),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),product_snd_int_int)),zero_zero_int))))),hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,cOMBS_480885903t_bool),hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int)),hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,cOMBB_47643171nt_int),hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,times_times_int),product_fst_int_int))),product_snd_int_int)))),hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,hAPP_f578362681nt_int(cOMBC_1417754532nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,times_times_int),product_fst_int_int))),product_snd_int_int)))))) ).

tff(fact_4851_divmod__nat__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( N = zero_zero_nat )
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = hAPP_P1586233937at_nat(produc1391996073at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,product_Pair_nat_nat),suc)),divmod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

tff(fact_4852_Bolzano__bisect_Osimps_I2_J,axiom,
    ! [P_1: fun_Pr1127657839l_bool,Aa: real,Ba: real,N: nat] : hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),hAPP_nat_nat(suc,N)) = hAPP_P1860904029l_real(produc595218619l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,cOMBS_902912708l_real),hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,cOMBS_794971624l_real),hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,hAPP_f1076489843l_real(cOMBB_415306361l_real,if_Pro313124157l_real)),hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,hAPP_f940061907l_bool(cOMBB_1795223523l_real,P_1)),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,cOMBS_511527878l_real),hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,hAPP_f2114902373l_real(cOMBB_237479429l_real,produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),cOMBI_real)))),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N)) ).

tff(fact_4853_suminf__exist__split,axiom,
    ! [Aa: fun_nat_real,R_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2))
     => ( hBOOL(summable_real(Aa))
       => ? [N_3: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,Aa),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_2))))),R_2)) ) ) ) ).

tff(fact_4854_divmod__int__pdivmod,axiom,
    ! [L: int,K_1: int] :
      ( ( ( K_1 = zero_zero_int )
       => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ) )
      & ( ( K_1 != zero_zero_int )
       => ( ( ( L = zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) ) )
          & ( ( L != zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fdisj(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)),hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,L),zero_zero_int)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) ) ) ) ) ) ).

tff(fact_4855_negateSnd__def,axiom,
    negateSnd = produc713050258nt_int(uminus_uminus_int) ).

tff(fact_4856_divmod__int__code,axiom,
    ! [L: int,K_1: int] :
      ( ( ( K_1 = zero_zero_int )
       => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ) )
      & ( ( K_1 != zero_zero_int )
       => ( ( ( L = zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) ) )
          & ( ( L != zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K_1)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) ) ) ) ) ) ).

tff(fact_4857_summable__le2,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2)))
     => ( hBOOL(summable_real(G))
       => ( hBOOL(summable_real(F))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) ) ) ) ).

tff(fact_4858_suminf__gt__zero,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_2)))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4859_suminf__ge__zero,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4860_suminf__0__le,axiom,
    ! [F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
     => ( hBOOL(summable_real(F))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4861_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,cOMBB_bool_bool_int),hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) ).

tff(fact_4862_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) ).

tff(fact_4863_sin__paired,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))),hAPP_real_real(sin,X_1))) ).

tff(fact_4864_sin__converges,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_real_real(sin,X_1))) ).

tff(fact_4865_cos__converges,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_real_real(cos,X_1))) ).

tff(fact_4866_power__half__series,axiom,
    hBOOL(hAPP_real_bool(sums_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),suc)),one_one_real)) ).

tff(fact_4867_Ln_Oaux2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4868_sums__if_H,axiom,
    ! [G: fun_nat_real,X_1: real] :
      ( hBOOL(hAPP_real_bool(sums_real(G),X_1))
     => hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),X_1)) ) ).

tff(fact_4869_sums__if,axiom,
    ! [F: fun_nat_real,Y_2: real,G: fun_nat_real,X_1: real] :
      ( hBOOL(hAPP_real_bool(sums_real(G),X_1))
     => ( hBOOL(hAPP_real_bool(sums_real(F),Y_2))
       => hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2))) ) ) ).

tff(fact_4870_cos__paired,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(cos,X_1))) ).

tff(fact_4871_cnj_Osums,axiom,
    ! [X_3: fun_nat_complex,Aa: complex] :
      ( hBOOL(hAPP_complex_bool(sums_complex(X_3),Aa))
     => hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)),hAPP_complex_complex(cnj,Aa))) ) ).

tff(fact_4872_Re_Osums,axiom,
    ! [X_3: fun_nat_complex,Aa: complex] :
      ( hBOOL(hAPP_complex_bool(sums_complex(X_3),Aa))
     => hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)),hAPP_complex_real(re,Aa))) ) ).

tff(fact_4873_Im_Osums,axiom,
    ! [X_3: fun_nat_complex,Aa: complex] :
      ( hBOOL(hAPP_complex_bool(sums_complex(X_3),Aa))
     => hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)),hAPP_complex_real(im,Aa))) ) ).

tff(fact_4874_sin__fdiffs2,axiom,
    ! [N: nat] : hAPP_nat_real(diffs_real(sin_coeff),N) = hAPP_nat_real(cos_coeff,N) ).

tff(fact_4875_sin__fdiffs,axiom,
    diffs_real(sin_coeff) = cos_coeff ).

tff(fact_4876_exp__fdiffs,axiom,
    ! [X_2: nat] : hAPP_nat_real(diffs_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),X_2) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2))) ).

tff(fact_4877_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
    <=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,M)),N)) ) ).

tff(fact_4878_cos__fdiffs,axiom,
    ! [X_2: nat] : hAPP_nat_real(diffs_real(cos_coeff),X_2) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X_2)) ).

tff(fact_4879_cos__fdiffs2,axiom,
    ! [N: nat] : hAPP_nat_real(diffs_real(cos_coeff),N) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,N)) ).

tff(fact_4880_diff__Suc,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

tff(fact_4881_summable__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(G,N_2)))
     => ( hBOOL(summable_real(F))
       => ( hBOOL(summable_real(G))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) ) ) ) ).

tff(fact_4882_arctan__def,axiom,
    ! [Y_2: real] : hAPP_real_real(arctan,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),tan)),Y_2)))) ).

tff(fact_4883_arcsin__def,axiom,
    ! [Y_2: real] : hAPP_real_real(arcsin,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),sin)),Y_2)))) ).

tff(fact_4884_ln__def,axiom,
    ! [X_1: real] : hAPP_real_real(ln,X_1) = the_real(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),exp_real)),X_1)) ).

tff(fact_4885_root__def,axiom,
    ! [N: nat,X_1: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_1) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N))),X_1))) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
           => ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_1) = hAPP_real_real(uminus_uminus_real,the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N))),hAPP_real_real(uminus_uminus_real,X_1))))) ) )
          & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
           => ( hAPP_real_real(hAPP_n546711566l_real(root,N),X_1) = zero_zero_real ) ) ) ) ) ).

tff(fact_4886_arccos__def,axiom,
    ! [Y_2: real] : hAPP_real_real(arccos,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),Y_2)))) ).

tff(fact_4887_pi__def,axiom,
    pi = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),zero_zero_real))))) ).

tff(fact_4888_pi__half,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),zero_zero_real)))) ).

tff(fact_4889_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = the_Pr588456374at_nat(divmod_nat_rel(M,N)) ).

tff(fact_4890_floor__real__def,axiom,
    ! [X_1: real] : archim1246769320r_real(X_1) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,ord_less_eq_real),ring_1_of_int_real)),X_1))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,X_1)),hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,ring_1_of_int_real),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) ).

tff(fact_4891_floor__rat__def,axiom,
    ! [X_1: rat] : archim791455193or_rat(X_1) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,ord_less_eq_rat),ring_1_of_int_rat)),X_1))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,X_1)),hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,ring_1_of_int_rat),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) ).

tff(fact_4892_RRset2norRR__def,axiom,
    ! [Aa: int,A: fun_int_bool,M: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
          & hBOOL(hAPP_int_bool(is_RRset(A),M))
          & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A)) )
       => ( hAPP_int_int(rRset2norRR(A,M),Aa) = hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(Aa)),M))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,M)))) ) )
      & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
            & hBOOL(hAPP_int_bool(is_RRset(A),M))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A)) )
       => ( hAPP_int_int(rRset2norRR(A,M),Aa) = zero_zero_int ) ) ) ).

tff(fact_4893_aux__some,axiom,
    ! [Aa: int,A: fun_int_bool,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(is_RRset(A),M))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Aa),hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(Aa)),M))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,M))))),M))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(Aa)),M))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,M))))),hAPP_i1948725293t_bool(norRRset,M))) ) ) ) ) ).

tff(fact_4894_Maclaurin__sin__bound,axiom,
    ! [X_1: real,N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,X_1)),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,X_1)),N)))) ).

tff(fact_4895_finite__less__ub,axiom,
    ! [U: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))) ) ).

tff(fact_4896_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U))) ).

tff(fact_4897_sumr__diff__mult__const,axiom,
    ! [F: fun_nat_real,N: nat,R_2: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),R_2)) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),F)),R_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) ).

tff(fact_4898_sumr__offset4,axiom,
    ! [K_1: nat,N_2: nat,F_2: fun_nat_real] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_1))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) ).

tff(fact_4899_sumr__offset2,axiom,
    ! [K_1: nat,N: nat,F_2: fun_nat_real] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) ).

tff(fact_4900_sumr__one__lb__realpow__zero,axiom,
    ! [F: fun_nat_real,N: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,zero_zero_nat)),N)) = zero_zero_real ).

tff(fact_4901_all__nat__less__eq,axiom,
    ! [P_1: fun_nat_bool,N: nat] :
      ( ! [M_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
         => hBOOL(hAPP_nat_bool(P_1,M_2)) )
    <=> ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)))
         => hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_4902_ex__nat__less__eq,axiom,
    ! [P_1: fun_nat_bool,N: nat] :
      ( ? [M_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
          & hBOOL(hAPP_nat_bool(P_1,M_2)) )
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)))
          & hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_4903_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),L) ).

tff(fact_4904_sumr__geometric,axiom,
    ! [N: nat,X_1: real] :
      ( ( X_1 != one_one_real )
     => ( hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real)) ) ) ).

tff(fact_4905_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,L)),U) = hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U) ).

tff(fact_4906_image__add__atLeastLessThan,axiom,
    ! [K_1: nat,I_1: nat,J: nat] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).

tff(fact_4907_image__Suc__atLeastLessThan,axiom,
    ! [I_1: nat,J: nat] : image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) ).

tff(fact_4908_Maclaurin__zero,axiom,
    ! [Diff: fun_na182379510l_real,N: nat,X_1: real] :
      ( ( X_1 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N)) = hAPP_real_real(hAPP_n546711566l_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

tff(fact_4909_subset__card__intvl__is__intvl,axiom,
    ! [A: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A)))))
     => ( A = hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ) ).

tff(fact_4910_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),F)),G)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))) ).

tff(fact_4911_sumr__minus__one__realpow__zero,axiom,
    ! [N: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),suc)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) = zero_zero_real ).

tff(fact_4912_lemma__STAR__cos2,axiom,
    ! [N: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,zero_zero_real)))),zero_zero_real)),hAPP_n1699378549t_bool(ord_at4362885an_nat(one_one_nat),N)) = zero_zero_real ).

tff(fact_4913_sumr__cos__zero__one,axiom,
    ! [N: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(suc,N))) = one_one_real ).

tff(fact_4914_Maclaurin__sin__expansion2,axiom,
    ! [N: nat,X_1: real] :
    ? [T_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
      & ( hAPP_real_real(sin,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ).

tff(fact_4915_Maclaurin__cos__expansion,axiom,
    ! [N: nat,X_1: real] :
    ? [T_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
      & ( hAPP_real_real(cos,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ).

tff(fact_4916_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U))) ).

tff(fact_4917_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U))) ).

tff(fact_4918_Sup__atLeastLessThan,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
     => ( comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Y_2),X_1)) = X_1 ) ) ).

tff(fact_4919_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)) = hAPP_int_nat(nat_1,U) ).

tff(fact_4920_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)) ).

tff(fact_4921_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : hAPP_i1948725293t_bool(ord_at641636577an_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U) ).

tff(fact_4922_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),L),hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L))) = hAPP_i1948725293t_bool(ord_at641636577an_int(L),U) ).

tff(fact_4923_Maclaurin__sin__expansion,axiom,
    ! [X_1: real,N: nat] :
    ? [T_1: real] : hAPP_real_real(sin,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ).

tff(fact_4924_Maclaurin__lemma2,axiom,
    ! [B: real,K_1: nat,Diff: fun_na182379510l_real,H: real,N: nat] :
      ( ! [M_2: nat,T_1: real] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T_1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H)) )
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) )
     => ( ( N = hAPP_nat_nat(suc,K_1) )
       => ! [M_2: nat,T_1: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_n546711566l_real(Diff,M_2))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,M_2))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),power_power_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_2))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B)),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_2)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_2))))))),T_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_2)))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,T_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_2))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,T_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_2)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,M_2)))))))))) ) ) ) ).

tff(fact_4925_Maclaurin__exp__lt,axiom,
    ! [N: nat,X_1: real] :
      ( ( X_1 != zero_zero_real )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T_1)))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
            & ( hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ) ) ).

tff(fact_4926_Maclaurin__exp__le,axiom,
    ! [N: nat,X_1: real] :
    ? [T_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
      & ( hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ).

tff(fact_4927_sumr__pos__lt__pair,axiom,
    ! [K_1: nat,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [D_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)))),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)),one_one_nat))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4928_Maclaurin__lemma,axiom,
    ! [F: fun_real_real,J: fun_nat_real,N: nat,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
     => ? [B_5: real] : hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),J)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_5),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) ) ).

tff(fact_4929_series__pos__less,axiom,
    ! [N: nat,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [M_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,M_2))) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4930_series__pos__le,axiom,
    ! [N: nat,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [M_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,M_2))) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4931_Maclaurin__all__lt__objl,axiom,
    ! [N: nat,X_1: real,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_2: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),X_2)))
        & ( X_1 != zero_zero_real )
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) )
     => ? [T_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T_1)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
          & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ) ).

tff(fact_4932_Maclaurin__all__lt,axiom,
    ! [X_1: real,N: nat,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X_1 != zero_zero_real )
         => ( ! [M_2: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),X_2)))
           => ? [T_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T_1)))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
                & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ) ) ) ) ).

tff(fact_4933_Maclaurin__bi__le,axiom,
    ! [X_1: real,N: nat,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
     => ( ! [M_2: nat,T_1: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1))) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) )
       => ? [T_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
            & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ) ) ).

tff(fact_4934_Maclaurin__minus,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N: nat,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
         => ( ! [M_2: nat,T_1: real] :
                ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),zero_zero_real)) )
               => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) )
           => ? [T_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T_1),zero_zero_real))
                & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ) ) ) ).

tff(fact_4935_Maclaurin2,axiom,
    ! [N: nat,Diff: fun_na182379510l_real,F: fun_real_real,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
     => ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
       => ( ! [M_2: nat,T_1: real] :
              ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H)) )
             => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) )
         => ? [T_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H))
              & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ) ) ).

tff(fact_4936_Maclaurin__objl,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N: nat,H: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
        & ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_2: nat,T_1: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) ) )
     => ? [T_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T_1))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T_1),H))
          & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ).

tff(fact_4937_Maclaurin2__objl,axiom,
    ! [N: nat,Diff: fun_na182379510l_real,F: fun_real_real,H: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
        & ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_2: nat,T_1: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) ) )
     => ? [T_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T_1))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H))
          & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ).

tff(fact_4938_Maclaurin,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N: nat,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
         => ( ! [M_2: nat,T_1: real] :
                ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),H)) )
               => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) )
           => ? [T_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T_1),H))
                & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ) ) ) ).

tff(fact_4939_Maclaurin__minus__objl,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N: nat,H: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
        & ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_2: nat,T_1: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T_1),zero_zero_real)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),T_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),T_1))) ) )
     => ? [T_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T_1))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T_1),zero_zero_real))
          & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N))) ) ) ) ).

tff(fact_4940_Maclaurin__all__le__objl,axiom,
    ! [N: nat,X_1: real,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_2: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),X_2))) )
     => ? [T_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
          & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ) ).

tff(fact_4941_Maclaurin__all__le,axiom,
    ! [N: nat,X_1: real,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
     => ( ! [M_2: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_2),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_2)),X_2)))
       => ? [T_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T_1)),hAPP_real_real(abs_abs_real,X_1)))
            & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N),T_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N))) ) ) ) ) ).

tff(fact_4942_real__setsum__nat__ivl__bounded,axiom,
    ! [F: fun_nat_real,K_3: real,N: nat] :
      ( ! [P_4: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),N))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,P_4)),K_3)) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),K_3))) ) ).

tff(fact_4943_DERIV__sumr,axiom,
    ! [F: fun_na182379510l_real,X_1: real,F_1: fun_na182379510l_real,N: nat,M: nat] :
      ( ! [R_1: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),R_1))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) )
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),X_1),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),X_1))) )
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f42270917t_real(cOMBC_nat_real_real,F))),hAPP_n1699378549t_bool(ord_at4362885an_nat(M),N)),X_1),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),X_1)),hAPP_n1699378549t_bool(ord_at4362885an_nat(M),N)))) ) ).

tff(fact_4944_suminf__le,axiom,
    ! [X_1: real,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_1))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),X_1)) ) ) ).

tff(fact_4945_transfer__nat__int__sum__prod__closure_I1_J,axiom,
    ! [F: fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_2))) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A))) ) ) ).

tff(fact_4946_pos__summable,axiom,
    ! [X_1: real,F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_1))
       => hBOOL(summable_real(F)) ) ) ).

tff(fact_4947_arith__series__int,axiom,
    ! [Aa: int,D: int,N: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,hAPP_int_fun_int_int(plus_plus_int,Aa)),hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,times_times_int),semiri1621563631at_int)),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),D)))) ).

tff(fact_4948_BseqI2,axiom,
    ! [K_3: real,K_1: real,F: fun_nat_real] :
      ( ! [N_2: nat] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K_1),hAPP_nat_real(F,N_2)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),K_3)) )
     => hBOOL(bseq_real(F)) ) ).

tff(fact_4949_finite__lessThan,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ).

tff(fact_4950_card__lessThan,axiom,
    ! [U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U ).

tff(fact_4951_atLeast0LessThan,axiom,
    ! [N: nat] : hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N) = hAPP_n1699378549t_bool(ord_lessThan_nat,N) ).

tff(fact_4952_finite__nat__iff__bounded,axiom,
    ! [S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ? [K: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ) ).

tff(fact_4953_arith__series__nat,axiom,
    ! [Aa: nat,D: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(plus_plus_nat,Aa)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),D)))) ).

tff(fact_4954_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U))
     => ( hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_int_nat(nat_1,U))) ) ) ).

tff(fact_4955_finite__nat__bounded,axiom,
    ! [S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
     => ? [K: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ) ).

tff(fact_4956_sum__diff__distrib,axiom,
    ! [N: nat,Q_2: fun_nat_nat,P_1: fun_nat_nat] :
      ( ! [X_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_2,X_2)),hAPP_nat_nat(P_1,X_2)))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,P_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_2),hAPP_n1699378549t_bool(ord_lessThan_nat,N))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),P_1)),Q_2)),hAPP_n1699378549t_bool(ord_lessThan_nat,N)) ) ) ).

tff(fact_4957_Sup__lessThan,axiom,
    ! [X_1: real] : comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,X_1)) = X_1 ).

tff(fact_4958_transfer__nat__int__set__cong,axiom,
    ! [P_2: fun_int_bool,P_1: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => ( hBOOL(hAPP_int_bool(P_1,X_2))
          <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
     => ( collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_2)) ) ) ).

tff(fact_4959_normalize__def,axiom,
    ! [P_3: product_prod_int_int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))
       => ( normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))
       => ( ( ( hAPP_P1175774780nt_int(product_snd_int_int,P_3) = zero_zero_int )
           => ( normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
          & ( ( hAPP_P1175774780nt_int(product_snd_int_int,P_3) != zero_zero_int )
           => ( normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))) ) ) ) ) ) ).

tff(fact_4960_gcd__dvd1__int,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),X)) ).

tff(fact_4961_gcd__dvd2__int,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),Y)) ).

tff(fact_4962_coprime__exp2__int,axiom,
    ! [N_1: nat,M_1: nat,A_2: int,B_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),M_1)) = one_one_int ) ) ).

tff(fact_4963_invertible__coprime__int,axiom,
    ! [X: int,Y: int,M_1: int] :
      ( ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M_1) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),M_1) = one_one_int ) ) ).

tff(fact_4964_coprime__dvd__mult__int,axiom,
    ! [M_1: int,K_2: int,N_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N_1) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1)) ) ) ).

tff(fact_4965_coprime__dvd__mult__iff__int,axiom,
    ! [M: int,K_1: int,N: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),N) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M)) ) ) ).

tff(fact_4966_divides__mult__int,axiom,
    ! [N_1: int,M_1: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),R_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),R_3))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1) = one_one_int )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)),R_3)) ) ) ) ).

tff(fact_4967_gcd__coprime__int,axiom,
    ! [B_4: int,A_4: int,A_2: int,B_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1) != zero_zero_int )
     => ( ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_4),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)) )
       => ( ( B_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_4),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)) )
         => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),B_4) = one_one_int ) ) ) ) ).

tff(fact_4968_transfer__int__nat__gcd_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).

tff(fact_4969_gcd__add__mult__int,axiom,
    ! [M_1: int,K_2: int,N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1) ).

tff(fact_4970_gcd__add2__int,axiom,
    ! [M_1: int,N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_1),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1) ).

tff(fact_4971_gcd__add1__int,axiom,
    ! [M_1: int,N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_1),N_1)),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1) ).

tff(fact_4972_coprime__minus__one__int,axiom,
    ! [N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int)),N_1) = one_one_int ).

tff(fact_4973_coprime__plus__one__int,axiom,
    ! [N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)),N_1) = one_one_int ).

tff(fact_4974_coprime__exp__int,axiom,
    ! [N_1: nat,D_3: int,A_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_2) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)) = one_one_int ) ) ).

tff(fact_4975_coprime__divisors__nat,axiom,
    ! [E_1: int,B_1: int,D_3: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),A_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E_1),B_1))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1) = one_one_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),E_1) = one_one_int ) ) ) ) ).

tff(fact_4976_gcd__1__int,axiom,
    ! [M_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),one_one_int) = one_one_int ).

tff(fact_4977_gcd__zero__int,axiom,
    ! [M: int,N: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) = zero_zero_int )
    <=> ( ( M = zero_zero_int )
        & ( N = zero_zero_int ) ) ) ).

tff(fact_4978_gcd__red__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) ).

tff(fact_4979_gcd__greatest__iff__int,axiom,
    ! [K_1: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) ) ) ).

tff(fact_4980_gcd__greatest__int,axiom,
    ! [N_1: int,K_2: int,M_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1))) ) ) ).

tff(fact_4981_dvd__gcd__D1__int,axiom,
    ! [I: int,M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),M_1)) ) ).

tff(fact_4982_dvd__gcd__D2__int,axiom,
    ! [I: int,M_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N_1)) ) ).

tff(fact_4983_gcd__assoc__int,axiom,
    ! [A_2: int,B_1: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_1),C_1)) ).

tff(fact_4984_gcd__int_Oleft__commute,axiom,
    ! [B_1: int,A_2: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_1),C_1)) ).

tff(fact_4985_gcd__commute__int,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_1),A_2) ).

tff(fact_4986_gcd__exp__int,axiom,
    ! [A_2: int,N_1: nat,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_2),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),N_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)),N_1) ).

tff(fact_4987_gcd__mult__distrib__int,axiom,
    ! [K_2: int,M_1: int,N_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N_1)) ).

tff(fact_4988_abs__gcd__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4989_gcd__idem__int,axiom,
    ! [X: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),X) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_4990_gcd__abs__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) ).

tff(fact_4991_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(abs_abs_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4992_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4993_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(uminus_uminus_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4994_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4995_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,Y) ) ) ).

tff(fact_4996_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),Y))
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,X) ) ) ).

tff(fact_4997_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

tff(fact_4998_gcd__0__left__int,axiom,
    ! [X: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_4999_gcd__0__int,axiom,
    ! [X: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_5000_coprime__rmult__int,axiom,
    ! [D_3: int,A_2: int,B_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_1) = one_one_int ) ) ).

tff(fact_5001_coprime__lmult__int,axiom,
    ! [D_3: int,A_2: int,B_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_2) = one_one_int ) ) ).

tff(fact_5002_coprime__mult__int,axiom,
    ! [B_1: int,D_3: int,A_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_2) = one_one_int )
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_1) = one_one_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_2),B_1)) = one_one_int ) ) ) ).

tff(fact_5003_gcd__mult__cancel__int,axiom,
    ! [M_1: int,K_2: int,N_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N_1) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M_1)),N_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M_1),N_1) ) ) ).

tff(fact_5004_coprime__mul__eq__int,axiom,
    ! [D: int,Aa: int,Ba: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Aa),Ba)) = one_one_int )
    <=> ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),Aa) = one_one_int )
        & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),Ba) = one_one_int ) ) ) ).

tff(fact_5005_Fract__coprime,axiom,
    ! [A_2: int,B_1: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1))) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_1) ).

tff(fact_5006_div__gcd__coprime__int,axiom,
    ! [B_1: int,A_2: int] :
      ( ( ( A_2 != zero_zero_int )
        | ( B_1 != zero_zero_int ) )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1))) = one_one_int ) ) ).

tff(fact_5007_gcd__non__0__int,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) ) ) ).

tff(fact_5008_transfer__nat__int__gcd__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ).

tff(fact_5009_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) ).

tff(fact_5010_gcd__le1__int,axiom,
    ! [B_1: int,A_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)),A_2)) ) ).

tff(fact_5011_gcd__le2__int,axiom,
    ! [A_2: int,B_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)),B_1)) ) ).

tff(fact_5012_gcd__unique__int,axiom,
    ! [Ba: int,Aa: int,D: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Aa))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ba))
        & ! [E: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),Aa))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),Ba)) )
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D)) ) )
    <=> ( D = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Aa),Ba) ) ) ).

tff(fact_5013_gcd__cases__int,axiom,
    ! [P_1: fun_int_bool,Y_2: int,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
         => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_2))) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(uminus_uminus_int,Y_2)))) ) )
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),zero_zero_int))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),Y_2))) ) )
         => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),zero_zero_int))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),hAPP_int_int(uminus_uminus_int,Y_2)))) ) )
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_2))) ) ) ) ) ).

tff(fact_5014_coprime__crossproduct__int,axiom,
    ! [Ba: int,C: int,Aa: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Aa),D) = one_one_int )
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ba),C) = one_one_int )
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,Aa)),hAPP_int_int(abs_abs_int,C)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,Ba)),hAPP_int_int(abs_abs_int,D)) )
        <=> ( ( hAPP_int_int(abs_abs_int,Aa) = hAPP_int_int(abs_abs_int,Ba) )
            & ( hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D) ) ) ) ) ) ).

tff(fact_5015_gcd__code__int,axiom,
    ! [K_1: int,L: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) ).

tff(fact_5016_coprime__common__divisor__int,axiom,
    ! [X: int,A_2: int,B_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),A_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),B_1))
         => ( hAPP_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).

tff(fact_5017_quotient__of__coprime,axiom,
    ! [R_3: rat,P_3: int,Q_1: int] :
      ( ( quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int ) ) ).

tff(fact_5018_normalize__coprime,axiom,
    ! [R_3: product_prod_int_int,P_3: int,Q_1: int] :
      ( ( normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int ) ) ).

tff(fact_5019_coprime__int,axiom,
    ! [Aa: int,Ba: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Aa),Ba) = one_one_int )
    <=> ! [D_2: int] :
          ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_2))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),Aa))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),Ba)) )
        <=> ( D_2 = one_one_int ) ) ) ).

tff(fact_5020_transfer__nat__int__gcd_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ) ).

tff(fact_5021_gcd__int__def,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Y)))) ).

tff(fact_5022_normalize__stable,axiom,
    ! [P_3: int,Q_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1))
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int )
       => ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) ) ) ) ).

tff(fact_5023_quotient__of__def,axiom,
    ! [X_1: rat] : quotient_of(X_1) = the_Pr2103884470nt_int(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,hAPP_r2115560837t_bool(fequal_rat,X_1)),hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,fract),product_fst_int_int)),product_snd_int_int)))),hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)),product_snd_int_int))),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,gcd_gcd_int),product_fst_int_int)),product_snd_int_int))),one_one_int)))) ).

tff(fact_5024_quotient__of__unique,axiom,
    ! [R_3: rat] :
    ? [X_2: product_prod_int_int] :
      ( ( R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) )
      & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X_2)))
      & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) = one_one_int )
      & ! [Y_1: product_prod_int_int] :
          ( ( ( R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) )
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)))
            & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) = one_one_int ) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_5025_Rat__cases__nonzero,axiom,
    ! [Q_1: rat] :
      ( ! [A_1: int,B_3: int] :
          ( ( Q_1 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_3) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_3))
           => ( ( A_1 != zero_zero_int )
             => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_3) != one_one_int ) ) ) )
     => ( Q_1 = zero_zero_rat ) ) ).

tff(fact_5026_Rat__induct,axiom,
    ! [Qa: rat,P_1: fun_rat_bool] :
      ( ! [A_1: int,B_3: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_3))
         => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_3) = one_one_int )
           => hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_3))) ) )
     => hBOOL(hAPP_rat_bool(P_1,Qa)) ) ).

tff(fact_5027_Rat__cases,axiom,
    ! [Q_1: rat] :
      ~ ! [A_1: int,B_3: int] :
          ( ( Q_1 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_3) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_3))
           => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_3) != one_one_int ) ) ) ).

tff(fact_5028_card__setsum__aux,axiom,
    ! [N: nat,S: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S))
     => ( ! [X_2: fun_int_bool] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),S))
           => hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)) )
       => ( ! [X_2: fun_int_bool] :
              ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),S))
             => ( hAPP_f957591787ol_nat(finite_card_int,X_2) = N ) )
         => ( hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,finite_card_int),S) = hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,cOMBK_1643584600t_bool(N)),S) ) ) ) ) ).

tff(fact_5029_zOdd__def,axiom,
    zOdd = collect_int(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int),hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,cOMBB_int_bool_int),fequal_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))) ).

tff(fact_5030_Rats__eq__int__div__int,axiom,
    field_1210416355s_real = collect_real(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,fEx_int),hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int)),hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,cOMBC_1683390479t_bool),hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool)),hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj))),hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,cOMBB_95956018ol_int),hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,cOMBB_real_bool_int),fequal_real))),hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,cOMBB_real_real_int),hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,inverse_divide_real),real_int))),real_int)))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))))) ).

tff(fact_5031_zEven__def,axiom,
    zEven = collect_int(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int),hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,cOMBB_int_bool_int),fequal_int)),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_5032_summable__Leibniz_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(Aa))
       => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),sequentially)) ) ) ).

tff(fact_5033_LIMSEQ__neg__powr,axiom,
    ! [S_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),S_1))
     => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,powr),real_nat)),hAPP_real_real(uminus_uminus_real,S_1)),zero_zero_real,sequentially)) ) ).

tff(fact_5034_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R_2: real] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),R_2,sequentially)) ).

tff(fact_5035_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,X_1)),zero_zero_real,sequentially)) ) ).

tff(fact_5036_LIMSEQ__divide__realpow__zero,axiom,
    ! [Aa: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,Aa)),hAPP_r474017924t_real(power_power_real,X_1)),zero_zero_real,sequentially)) ) ).

tff(fact_5037_LIMSEQ__inverse__real__of__nat,axiom,
    hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)),zero_zero_real,sequentially)) ).

tff(fact_5038_LIMSEQ__rabs__zero,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially))
    <=> hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)) ) ).

tff(fact_5039_LIMSEQ__imp__rabs,axiom,
    ! [F: fun_nat_real,L: real] :
      ( hBOOL(tendsto_nat_real(F,L,sequentially))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),hAPP_real_real(abs_abs_real,L),sequentially)) ) ).

tff(fact_5040_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R_2: real] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))),R_2,sequentially)) ).

tff(fact_5041_LIMSEQ__rabs__realpow__zero2,axiom,
    ! [C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real))
     => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,C),zero_zero_real,sequentially)) ) ).

tff(fact_5042_LIMSEQ__rabs__realpow__zero,axiom,
    ! [C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real))
     => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,C)),zero_zero_real,sequentially)) ) ).

tff(fact_5043_monoseq__le,axiom,
    ! [X_1: real,Aa: fun_nat_real] :
      ( hBOOL(monoseq_real(Aa))
     => ( hBOOL(tendsto_nat_real(Aa,X_1,sequentially))
       => ( ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,N_2)),X_1))
            & ! [M_2: nat,N_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,M_2)),hAPP_nat_real(Aa,N_2))) ) )
          | ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(Aa,N_2)))
            & ! [M_2: nat,N_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,N_2)),hAPP_nat_real(Aa,M_2))) ) ) ) ) ) ).

tff(fact_5044_LIMSEQ__le,axiom,
    ! [Y_3: fun_nat_real,Y_2: real,X_3: fun_nat_real,X_1: real] :
      ( hBOOL(tendsto_nat_real(X_3,X_1,sequentially))
     => ( hBOOL(tendsto_nat_real(Y_3,Y_2,sequentially))
       => ( ? [N_3: nat] :
            ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),hAPP_nat_real(Y_3,N_2))) )
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ) ).

tff(fact_5045_LIMSEQ__le__const2,axiom,
    ! [Aa: real,X_3: fun_nat_real,X_1: real] :
      ( hBOOL(tendsto_nat_real(X_3,X_1,sequentially))
     => ( ? [N_3: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),Aa)) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Aa)) ) ) ).

tff(fact_5046_LIMSEQ__le__const,axiom,
    ! [Aa: real,X_3: fun_nat_real,X_1: real] :
      ( hBOOL(tendsto_nat_real(X_3,X_1,sequentially))
     => ( ? [N_3: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),hAPP_nat_real(X_3,N_2))) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_1)) ) ) ).

tff(fact_5047_LIMSEQ__realpow__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,X_1),zero_zero_real,sequentially)) ) ) ).

tff(fact_5048_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R_2: real] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,one_one_real)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))))),R_2,sequentially)) ).

tff(fact_5049_summable__Leibniz_I1_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(Aa))
       => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))) ) ) ).

tff(fact_5050_zeroseq__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),zero_zero_real,sequentially)) ) ).

tff(fact_5051_Rats__eq__int__div__nat,axiom,
    field_1210416355s_real = collect_real(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,fEx_int),hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,hAPP_f389116883t_bool(cOMBB_1593174431ol_int,fEx_nat)),hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,cOMBC_678732631t_bool),hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,cOMBS_nat_bool_bool)),hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,hAPP_f643658017l_bool(cOMBB_2046938128ol_int,hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj))),hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,cOMBB_1647910634ol_int),hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,cOMBB_real_bool_nat),fequal_real))),hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,cOMBB_real_real_nat),hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,inverse_divide_real),real_int))),real_nat)))))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),zero_zero_nat)))))) ).

tff(fact_5052_summable__Leibniz_I4_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(Aa))
       => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),sequentially)) ) ) ).

tff(fact_5053_summable__Leibniz_H_I4_J,axiom,
    ! [N: nat,Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(Aa,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(Aa,N_2)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),one_one_nat))))) ) ) ) ).

tff(fact_5054_summable__Leibniz_H_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(Aa,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(Aa,N_2)))
         => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),sequentially)) ) ) ) ).

tff(fact_5055_cnj_OLIMSEQ,axiom,
    ! [X_3: fun_nat_complex,Aa: complex] :
      ( hBOOL(tendsto_nat_complex(X_3,Aa,sequentially))
     => hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3),hAPP_complex_complex(cnj,Aa),sequentially)) ) ).

tff(fact_5056_Re_OLIMSEQ,axiom,
    ! [X_3: fun_nat_complex,Aa: complex] :
      ( hBOOL(tendsto_nat_complex(X_3,Aa,sequentially))
     => hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3),hAPP_complex_real(re,Aa),sequentially)) ) ).

tff(fact_5057_Im_OLIMSEQ,axiom,
    ! [X_3: fun_nat_complex,Aa: complex] :
      ( hBOOL(tendsto_nat_complex(X_3,Aa,sequentially))
     => hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3),hAPP_complex_real(im,Aa),sequentially)) ) ).

tff(fact_5058_LIMSEQ__Complex,axiom,
    ! [Y_3: fun_nat_real,Ba: real,X_3: fun_nat_real,Aa: real] :
      ( hBOOL(tendsto_nat_real(X_3,Aa,sequentially))
     => ( hBOOL(tendsto_nat_real(Y_3,Ba,sequentially))
       => hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex_1),X_3)),Y_3),hAPP_real_complex(hAPP_r265291036omplex(complex_1,Aa),Ba),sequentially)) ) ) ).

tff(fact_5059_Bolzano__nest__unique,axiom,
    ! [P_1: fun_Pr1127657839l_bool,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
         => ( hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,Aa,Ba)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,Aa,Ba))),zero_zero_real,sequentially))
           => ? [L_1: real] :
                ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2))),L_1))
                & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,Aa,Ba)),L_1,sequentially))
                & ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L_1),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,Aa,Ba),N_2))))
                & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,Aa,Ba)),L_1,sequentially)) ) ) ) ) ) ).

tff(fact_5060_summable__Leibniz_H_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(Aa,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(Aa,N_2)))
         => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),sequentially)) ) ) ) ).

tff(fact_5061_summable__Leibniz_H_I2_J,axiom,
    ! [N: nat,Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(Aa,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(Aa,N_2)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)))) ) ) ) ).

tff(fact_5062_summable__Leibniz_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(Aa))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(Aa,zero_zero_nat)),zero_zero_real))
         => ! [N_2: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))))) ) ) ) ).

tff(fact_5063_summable__Leibniz_I2_J,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(Aa))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(Aa,zero_zero_nat)))
         => ! [N_2: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))))) ) ) ) ).

tff(fact_5064_Sup__atLeastAtMost,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1))
     => ( comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Y_2),X_1)) = X_1 ) ) ).

tff(fact_5065_summable,axiom,
    ! [Aa: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(Aa,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(Aa,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(Aa,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(Aa,N_2)))
         => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),Aa))) ) ) ) ).

tff(fact_5066_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U))) ).

tff(fact_5067_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) ).

tff(fact_5068_ex__nat__less,axiom,
    ! [P_1: fun_nat_bool,N: nat] :
      ( ? [M_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N))
          & hBOOL(hAPP_nat_bool(P_1,M_2)) )
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N)))
          & hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_5069_all__nat__less,axiom,
    ! [P_1: fun_nat_bool,N: nat] :
      ( ! [M_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N))
         => hBOOL(hAPP_nat_bool(P_1,M_2)) )
    <=> ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N)))
         => hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_5070_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_1),Y_2))) ) ).

tff(fact_5071_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hAPP_n1699378549t_bool(ord_at4362885an_nat(L),hAPP_nat_nat(suc,U)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U) ).

tff(fact_5072_fact__altdef__nat,axiom,
    ! [N: nat] : hAPP_nat_nat(fact_fact_nat,N) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),N)) ).

tff(fact_5073_image__add__atLeastAtMost,axiom,
    ! [K_1: nat,I_1: nat,J: nat] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).

tff(fact_5074_image__Suc__atLeastAtMost,axiom,
    ! [I_1: nat,J: nat] : image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) ).

tff(fact_5075_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
    ! [M: nat,N: nat] : hAPP_n1699378549t_bool(ord_at238088361st_nat(M),N) = image_int_nat(nat_1,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N))) ).

tff(fact_5076_cnj_Ocont,axiom,
    ! [Aa: complex] : hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,Aa),at_complex(Aa))) ).

tff(fact_5077_Re_Ocont,axiom,
    ! [Aa: complex] : hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,Aa),at_complex(Aa))) ).

tff(fact_5078_Im_Ocont,axiom,
    ! [Aa: complex] : hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,Aa),at_complex(Aa))) ).

tff(fact_5079_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,U)),L) ).

tff(fact_5080_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hAPP_i1948725293t_bool(ord_at641636577an_int(L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),one_one_int)) = hAPP_i1948725293t_bool(ord_at875362053st_int(L),U) ).

tff(fact_5081_setsum__shift__lb__Suc0__0,axiom,
    ! [K_1: nat,F: fun_nat_nat] :
      ( ( hAPP_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,zero_zero_nat)),K_1)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K_1)) ) ) ).

tff(fact_5082_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

tff(fact_5083_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N)) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

tff(fact_5084_fact__altdef__int,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_int_int(fact_fact_int,N) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),N)) ) ) ).

tff(fact_5085_LIM__cos__div__sin,axiom,
    hBOOL(tendsto_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),cos)),sin),zero_zero_real,at_real(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_5086_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( hBOOL(tendsto_real_real(F,L,at_real(C)))
     => ( ( L != zero_zero_real )
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ! [X_2: real] :
                ( ( ( X_2 != C )
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) )
               => ( hAPP_real_real(F,X_2) != zero_zero_real ) ) ) ) ) ).

tff(fact_5087_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( hBOOL(tendsto_real_real(F,L,at_real(C)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ! [X_2: real] :
                ( ( ( X_2 != C )
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) )
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X_2))) ) ) ) ) ).

tff(fact_5088_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( hBOOL(tendsto_real_real(F,L,at_real(C)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ! [X_2: real] :
                ( ( ( X_2 != C )
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) )
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),zero_zero_real)) ) ) ) ) ).

tff(fact_5089_aset_I8_J,axiom,
    ! [T: int,A: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) ) ) ) ).

tff(fact_5090_aset_I6_J,axiom,
    ! [T: int,A: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,T),one_one_int)),A))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),T))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),T)) ) ) ) ) ).

tff(fact_5091_bset_I8_J,axiom,
    ! [T: int,B: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T),one_one_int)),B))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T),X_2))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) ) ) ) ) ).

tff(fact_5092_bset_I6_J,axiom,
    ! [T: int,B: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),T))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),T)) ) ) ) ).

tff(fact_5093_bset_I3_J,axiom,
    ! [T: int,B: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T),one_one_int)),B))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( ( X_2 = T )
             => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1) = T ) ) ) ) ) ).

tff(fact_5094_aset_I3_J,axiom,
    ! [T: int,A: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,T),one_one_int)),A))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( ( X_2 = T )
             => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1) = T ) ) ) ) ) ).

tff(fact_5095_aset_I7_J,axiom,
    ! [T: int,A: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,T),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,T),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) ) ) ) ).

tff(fact_5096_bset_I5_J,axiom,
    ! [T: int,B: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),T))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),T)) ) ) ) ).

tff(fact_5097_aset_I5_J,axiom,
    ! [T: int,A: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,T),A))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),T))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),T)) ) ) ) ) ).

tff(fact_5098_bset_I7_J,axiom,
    ! [T: int,B: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,T),B))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,T),X_2))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,T),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) ) ) ) ) ).

tff(fact_5099_aset_I4_J,axiom,
    ! [T: int,A: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,T),A))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( ( X_2 != T )
             => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1) != T ) ) ) ) ) ).

tff(fact_5100_bset_I4_J,axiom,
    ! [T: int,B: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,T),B))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( ( X_2 != T )
             => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1) != T ) ) ) ) ) ).

tff(fact_5101_periodic__finite__ex,axiom,
    ! [P_1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int,K: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
          <=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
        <=> ? [X_2: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
              & hBOOL(hAPP_int_bool(P_1,X_2)) ) ) ) ) ).

tff(fact_5102_aset_I10_J,axiom,
    ! [T: int,A: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),T)))
           => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),T))) ) ) ) ).

tff(fact_5103_bset_I10_J,axiom,
    ! [T: int,B: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),T)))
           => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),T))) ) ) ) ).

tff(fact_5104_bset_I9_J,axiom,
    ! [T: int,B: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),T)))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),T))) ) ) ) ).

tff(fact_5105_aset_I9_J,axiom,
    ! [T: int,A: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),T)))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),T))) ) ) ) ).

tff(fact_5106_LIMSEQ__inverse__zero,axiom,
    ! [X_3: fun_nat_real] :
      ( ! [R_1: real] :
        ? [N_3: nat] :
        ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_3,N_2))) )
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_3),zero_zero_real,sequentially)) ) ).

tff(fact_5107_cnj_OscaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_complex_complex(cnj,X)) ).

tff(fact_5108_Re_OscaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(re,X)) ).

tff(fact_5109_Im_OscaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(im,X)) ).

tff(fact_5110_real__scaleR__def,axiom,
    ! [A_2: real,X: real] : hAPP_real_real(scaleR_scaleR_real(A_2),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_2),X) ).

tff(fact_5111_complex__scaleR,axiom,
    ! [R_3: real,A_2: real,B_1: real] : hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_2),B_1)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),B_1)) ).

tff(fact_5112_complex__Im__scaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X)) ).

tff(fact_5113_complex__Re__scaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X)) ).

tff(fact_5114_complex__sgn__def,axiom,
    ! [X: complex] : hAPP_complex_complex(sgn_sgn_complex,X) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X))),X) ).

tff(fact_5115_complex__scaleR__def,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_complex(scaleR1652505878omplex(R_3),X) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X))) ).

tff(fact_5116_trivial__limit__sequentially,axiom,
    ~ hBOOL(trivial_limit_nat(sequentially)) ).

tff(fact_5117_pred__nat__def,axiom,
    pred_nat = collec1979865426at_nat(produc1922581855t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat)),suc))) ).

tff(fact_5118_gcd__coprime__exists__int,axiom,
    ! [A_2: int,B_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1) != zero_zero_int )
     => ? [A_3: int,B_2: int] :
          ( ( A_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)) )
          & ( B_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_1)) )
          & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int ) ) ) ).

tff(fact_5119_min__Suc__number__of,axiom,
    ! [V: int,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N))) ).

tff(fact_5120_min__number__of__Suc,axiom,
    ! [N: nat,V: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) ).

tff(fact_5121_min__Suc__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M_1),N_1)) ).

tff(fact_5122_min__0R,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_1),zero_zero_nat) = zero_zero_nat ).

tff(fact_5123_min__0L,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N_1) = zero_zero_nat ).

tff(fact_5124_min__diff,axiom,
    ! [M_1: nat,I: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M_1),N_1)),I) ).

tff(fact_5125_min__Suc2,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_min_nat),N))),M) ).

tff(fact_5126_min__Suc1,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N)),M) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_min_nat,N))),M) ).

tff(fact_5127_vanishes__mult__bounded,axiom,
    ! [Y_3: fun_nat_rat,X_3: fun_nat_rat] :
      ( ? [A_1: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_1))
          & ! [N_2: nat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),A_1)) )
     => ( hBOOL(vanishes(Y_3))
       => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,times_times_rat),X_3)),Y_3))) ) ) ).

tff(fact_5128_SetInterval_Otransfer__int__nat__set__functions,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,M))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,N))
       => ( hAPP_i1948725293t_bool(ord_at875362053st_int(M),N) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat_1,M)),hAPP_int_nat(nat_1,N))) ) ) ) ).

tff(fact_5129_is__nat__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).

tff(fact_5130_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
    ! [Z_1: nat] : hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(semiri1621563631at_int,Z_1))) ).

tff(fact_5131_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_5132_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,one_one_int)) ).

tff(fact_5133_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,zero_zero_int)) ).

tff(fact_5134_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_int_bool(nat_is_nat,X_1)) ) ) ).

tff(fact_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
    ! [N_1: nat,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N_1))) ) ).

tff(fact_5136_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X),Y))) ) ) ).

tff(fact_5137_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))) ) ) ).

tff(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(nat_tsub(X),Y))) ) ) ).

tff(fact_5139_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_5140_vanishes__minus,axiom,
    ! [X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_3))) ) ).

tff(fact_5141_vanishes__const,axiom,
    ! [C: rat] :
      ( hBOOL(vanishes(cOMBK_rat_nat(C)))
    <=> ( C = zero_zero_rat ) ) ).

tff(fact_5142_vanishes__diff,axiom,
    ! [Y_3: fun_nat_rat,X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => ( hBOOL(vanishes(Y_3))
       => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,minus_minus_rat),X_3)),Y_3))) ) ) ).

tff(fact_5143_vanishes__add,axiom,
    ! [Y_3: fun_nat_rat,X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => ( hBOOL(vanishes(Y_3))
       => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,plus_plus_rat),X_3)),Y_3))) ) ) ).

tff(fact_5144_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
     => hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_1),Y_2))) ) ).

tff(fact_5145_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_5146_transfer__int__nat__gcd__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ).

tff(fact_5147_transfer__int__nat__factorial__closure,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))) ) ).

tff(fact_5148_transfer__int__nat__set__relations_I2_J,axiom,
    ! [A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
        <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat_1,X_1)),image_int_nat(nat_1,A))) ) ) ) ).

tff(fact_5149_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_5150_vanishesD,axiom,
    ! [R_2: rat,X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2))
       => ? [K: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_2)) ) ) ) ).

tff(fact_5151_vanishes__def,axiom,
    ! [X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
    <=> ! [R_1: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
         => ? [K: nat] :
            ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)) ) ) ) ).

tff(fact_5152_vanishesI,axiom,
    ! [X_3: fun_nat_rat] :
      ( ! [R_1: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
         => ? [K: nat] :
            ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)) ) )
     => hBOOL(vanishes(X_3)) ) ).

tff(fact_5153_zcongm__def,axiom,
    ! [M: int,X_2: int,Xa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(M),X_2),Xa))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),Xa),M)) ) ).

tff(fact_5154_cppi,axiom,
    ! [A: fun_int_bool,P_2: fun_int_bool,P_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ? [Z: int] :
          ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
           => ( hBOOL(hAPP_int_bool(P_1,X_2))
            <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
       => ( ! [X_2: int] :
              ( ! [Xa: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                 => ! [Xb: int] :
                      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
                     => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) ) )
         => ( ! [X_2: int,K: int] :
                ( hBOOL(hAPP_int_bool(P_2,X_2))
              <=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) )
           => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
            <=> ( ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & hBOOL(hAPP_int_bool(P_2,X_2)) )
                | ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & ? [Xa: int] :
                        ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A))
                        & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),X_2))) ) ) ) ) ) ) ) ) ).

tff(fact_5155_bijzcong__zcong__prod,axiom,
    ! [A: fun_int_bool,B: fun_int_bool,M: int] :
      ( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A),B)),bijR_int_int(zcongm(M))))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A)),big_co1548731110nt_int(cOMBI_int,B)),M)) ) ).

tff(fact_5156_cpmi,axiom,
    ! [B: fun_int_bool,P_2: fun_int_bool,P_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ? [Z: int] :
          ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
           => ( hBOOL(hAPP_int_bool(P_1,X_2))
            <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
       => ( ! [X_2: int] :
              ( ! [Xa: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                 => ! [Xb: int] :
                      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
                     => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) ) )
         => ( ! [X_2: int,K: int] :
                ( hBOOL(hAPP_int_bool(P_2,X_2))
              <=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) )
           => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
            <=> ( ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & hBOOL(hAPP_int_bool(P_2,X_2)) )
                | ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & ? [Xa: int] :
                        ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),B))
                        & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),X_2))) ) ) ) ) ) ) ) ) ).

tff(fact_5157_increasing__LIMSEQ,axiom,
    ! [L: real,F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),L))
       => ( ! [E: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
             => ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,N_2)),E))) )
         => hBOOL(tendsto_nat_real(F,L,sequentially)) ) ) ) ).

tff(fact_5158_isCont__real__root,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_1)) ) ).

tff(fact_5159_isCont__ln,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(isCont_real_real(ln),X_1)) ) ).

tff(fact_5160_isCont__abs,axiom,
    ! [Aa: real] : hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),Aa)) ).

tff(fact_5161_Im_OisCont,axiom,
    ! [Aa: complex] : hBOOL(hAPP_complex_bool(isCont_complex_real(im),Aa)) ).

tff(fact_5162_Re_OisCont,axiom,
    ! [Aa: complex] : hBOOL(hAPP_complex_bool(isCont_complex_real(re),Aa)) ).

tff(fact_5163_cnj_OisCont,axiom,
    ! [Aa: complex] : hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),Aa)) ).

tff(fact_5164_isCont__real__sqrt,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(sqrt),X_1)) ).

tff(fact_5165_isCont__cos,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(cos),X_1)) ).

tff(fact_5166_isCont__arctan,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(arctan),X_1)) ).

tff(fact_5167_isCont__sin,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(sin),X_1)) ).

tff(fact_5168_isCont__tan,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) != zero_zero_real )
     => hBOOL(hAPP_real_bool(isCont_real_real(tan),X_1)) ) ).

tff(fact_5169_isCont__root__zero,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),zero_zero_real)) ) ).

tff(fact_5170_isCont__root__neg,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
       => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_1)) ) ) ).

tff(fact_5171_isCont__root__pos,axiom,
    ! [X_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N)),X_1)) ) ) ).

tff(fact_5172_isCont__arccos,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(isCont_real_real(arccos),X_1)) ) ) ).

tff(fact_5173_isCont__arcsin,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(isCont_real_real(arcsin),X_1)) ) ) ).

tff(fact_5174_LIM__less__bound,axiom,
    ! [F: fun_real_real,Ba: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),X_1))
     => ( ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(Ba),X_1)))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_2))) )
       => ( hBOOL(hAPP_real_bool(isCont_real_real(F),X_1))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_1))) ) ) ) ).

tff(fact_5175_DERIV__inverse__function,axiom,
    ! [Ba: real,Aa: real,F: fun_real_real,G: fun_real_real,X_1: real,D_1: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,X_1)),D_1))
     => ( ( D_1 != zero_zero_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),X_1))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Ba))
           => ( ! [Y_1: real] :
                  ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Y_1))
                    & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),Ba)) )
                 => ( hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1 ) )
             => ( hBOOL(hAPP_real_bool(isCont_real_real(G),X_1))
               => hBOOL(hAPP_real_bool(deriv_real(G,X_1),hAPP_real_real(inverse_inverse_real,D_1))) ) ) ) ) ) ) ).

tff(fact_5176_isCont__inv__fun,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,X_1))) ) ) ) ).

tff(fact_5177_DERIV__isconst2,axiom,
    ! [X_1: real,F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) )
       => ( ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Ba)) )
             => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_1))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Ba))
             => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Aa) ) ) ) ) ) ) ).

tff(fact_5178_DERIV__isconst__end,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) )
       => ( ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Ba)) )
             => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
         => ( hAPP_real_real(F,Ba) = hAPP_real_real(F,Aa) ) ) ) ) ).

tff(fact_5179_DERIV__isconst1,axiom,
    ! [F: fun_real_real,Aa: real,Ba: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
           => hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) )
       => ( ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Ba)) )
             => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
         => ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Ba)) )
             => ( hAPP_real_real(F,X_2) = hAPP_real_real(F,Aa) ) ) ) ) ) ).

tff(fact_5180_lemma__isCont__inj2,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => ? [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,X_1))) ) ) ) ) ).

tff(fact_5181_isCont__inv__fun__inv,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => ? [E: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
              & ! [Y_1: real] :
                  ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,X_1)))))
                    & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,X_1)))),E)) )
                 => ( hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1 ) ) ) ) ) ) ).

tff(fact_5182_lemma__isCont__inj,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => ? [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,Z))) ) ) ) ) ).

tff(fact_5183_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [N: nat] : hAPP_n1699378549t_bool(ord_atMost_nat,N) = image_int_nat(nat_1,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).

tff(fact_5184_max__Suc__number__of,axiom,
    ! [V: int,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N))) ).

tff(fact_5185_finite__atMost,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K_1))) ).

tff(fact_5186_atLeast0AtMost,axiom,
    ! [N: nat] : hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N) = hAPP_n1699378549t_bool(ord_atMost_nat,N) ).

tff(fact_5187_card__atMost,axiom,
    ! [U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U)) = hAPP_nat_nat(suc,U) ).

tff(fact_5188_max__Suc__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M_1),N_1)) ).

tff(fact_5189_max__0R,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),zero_zero_nat) = N_1 ).

tff(fact_5190_max__0L,axiom,
    ! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N_1) = N_1 ).

tff(fact_5191_nat__minus__add__max,axiom,
    ! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1)),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),M_1) ).

tff(fact_5192_max__Suc1,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N)),M) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,N))),M) ).

tff(fact_5193_max__Suc2,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_max_nat),N))),M) ).

tff(fact_5194_lessThan__Suc__atMost,axiom,
    ! [K_1: nat] : hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K_1)) = hAPP_n1699378549t_bool(ord_atMost_nat,K_1) ).

tff(fact_5195_finite__nat__iff__bounded__le,axiom,
    ! [S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ? [K: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K))) ) ).

%----Helper facts (224)
tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fNot,P)) ) ).

tff(help_fTrue_1_1_U,axiom,
    hBOOL(fTrue) ).

tff(help_fTrue_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(P) ) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(Q) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | hBOOL(fdisj(P,Q)) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(fdisj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(fdisj(P,Q))
      | hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ hBOOL(fFalse) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fimplies(P),Q)) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(fimplies(P),Q)) ) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fimplies(P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_If_1_1_If_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__Int__Oint_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fEx_1_1_fEx_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_bool,X: int] :
      ( ~ hBOOL(hAPP_int_bool(P,X))
      | hBOOL(hAPP_f448129468l_bool(fEx_int,P)) ) ).

tff(help_fEx_1_1_fEx_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(P,X))
      | hBOOL(hAPP_f54304608l_bool(fEx_nat,P)) ) ).

tff(help_COMBI_1_1_COMBI_000tc__Int__Oint_U,axiom,
    ! [P: int] : hAPP_int_int(cOMBI_int,P) = P ).

tff(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
    ! [P: nat] : hAPP_nat_nat(cOMBI_nat,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)) ) ).

tff(help_If_1_1_If_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__RealDef__Oreal_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBI_1_1_COMBI_000tc__RealDef__Oreal_U,axiom,
    ! [P: real] : hAPP_real_real(cOMBI_real,P) = P ).

tff(help_fequal_1_1_fequal_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__Rat__Orat_000tc__Nat__Onat_U,axiom,
    ! [P: rat,Q: nat] : hAPP_nat_rat(cOMBK_rat_nat(P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: bool,Q: nat] : hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ).

tff(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_int,Q: fun_int_int,R: int] : hAPP_int_int(hAPP_f1356864277nt_int(hAPP_f747162069nt_int(cOMBB_int_int_int,P),Q),R) = hAPP_int_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_int,Q: fun_nat_int,R: nat] : hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R) = hAPP_int_int(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Rat__Orat_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_rat,Q: fun_int_int,R: int] : hAPP_int_rat(hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,P),Q),R) = hAPP_int_rat(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__Rat__Orat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_rat_rat,Q: fun_nat_rat,R: nat] : hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,P),Q),R) = hAPP_rat_rat(P,hAPP_nat_rat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_fun_int_int,Q: int,R: int] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R) = hAPP_int_int(hAPP_int_fun_int_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_fun_nat_int,Q: nat,R: int] : hAPP_int_int(hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,P),Q),R) = hAPP_nat_int(hAPP_int_fun_nat_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_nat_fun_int_int,Q: int,R: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,P),Q),R) = hAPP_int_int(hAPP_nat_fun_int_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: nat,R: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat_U,axiom,
    ! [P: fun_nat_fun_rat_rat,Q: fun_nat_rat,R: nat] : hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,P),Q),R) = hAPP_rat_rat(hAPP_nat_fun_rat_rat(P,R),hAPP_nat_rat(Q,R)) ).

tff(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
    ! [X: produc914805421l_real,Y: produc914805421l_real] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
    ! [X: produc914805421l_real,Y: produc914805421l_real] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_bool,Q: fun_int_int,R: int] : hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R) = hAPP_int_bool(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_bool,Q: fun_nat_int,R: nat] : hAPP_nat_bool(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P),Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_nat_bool,Q: fun_int_nat,R: int] : hAPP_int_bool(hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P),Q),R) = hAPP_nat_bool(P,hAPP_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_rat_bool,Q: fun_int_rat,R: int] : hAPP_int_bool(hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,P),Q),R) = hAPP_rat_bool(P,hAPP_int_rat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_int_bool,Q: int,R: int] : hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_rat_bool,Q: rat,R: int] : hAPP_int_bool(hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,P),Q),R) = hAPP_rat_bool(hAPP_i16292313t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_bool_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_in531499254l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_real,Q: fun_int_int,R: int] : hAPP_int_real(hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,P),Q),R) = hAPP_int_real(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_real,Q: fun_nat_nat,R: nat] : hAPP_nat_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,P),Q),R) = hAPP_nat_real(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_real_bool,Q: fun_int_real,R: int] : hAPP_int_bool(hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,P),Q),R) = hAPP_real_bool(P,hAPP_int_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_real_bool,Q: fun_nat_real,R: nat] : hAPP_nat_bool(hAPP_f737665877t_bool(hAPP_f605995867t_bool(cOMBB_real_bool_nat,P),Q),R) = hAPP_real_bool(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__HOL__Obool_U,axiom,
    ! [P: fun_in1251377736l_bool,Q: real,R: int] : hAPP_int_bool(hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,P),Q),R) = hAPP_real_bool(hAPP_i688218208l_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_re1114994977l_bool,Q: fun_real_bool,R: real] : hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Int__Oin,axiom,
    ! [P: fun_real_real,Q: fun_int_real,R: int] : hAPP_int_real(hAPP_f269654879t_real(hAPP_f766218899t_real(cOMBB_real_real_int,P),Q),R) = hAPP_real_real(P,hAPP_int_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona,axiom,
    ! [P: fun_real_real,Q: fun_nat_real,R: nat] : hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R) = hAPP_real_real(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea,axiom,
    ! [P: fun_na182379510l_real,Q: real,R: nat] : hAPP_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__Nat__Onat_000tc__RealDef__Orea,axiom,
    ! [P: fun_re1927296998t_real,Q: nat,R: real] : hAPP_real_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,P),Q),R) = hAPP_nat_real(hAPP_r474017924t_real(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: nat,Q: fun_int_bool] : hAPP_f957591787ol_nat(cOMBK_1643584600t_bool(P),Q) = P ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea,axiom,
    ! [P: fun_na182379510l_real,Q: fun_nat_real,R: nat] : hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__RealDef__Ore,axiom,
    ! [P: fun_real_bool,Q: fun_real_real,R: real] : hAPP_real_bool(hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,P),Q),R) = hAPP_real_bool(P,hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL__Obo,axiom,
    ! [P: fun_re413263731l_bool,Q: real,R: real] : hAPP_real_bool(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,P),Q),R) = hAPP_real_bool(hAPP_r1134773055l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__,axiom,
    ! [P: fun_complex_real,Q: fun_nat_complex,R: nat] : hAPP_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,P),Q),R) = hAPP_complex_real(P,hAPP_nat_complex(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocom,axiom,
    ! [P: fun_na1566252557omplex,Q: fun_nat_real,R: nat] : hAPP_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,P),Q),R) = hAPP_real_complex(hAPP_n1983812447omplex(P,R),hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
    ! [P: fun_real_real,Q: fun_real_real,R: real] : hAPP_real_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,P),Q),R) = hAPP_real_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
    ! [P: fun_re229243589l_real,Q: real,R: real] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
    ! [P: fun_re229243589l_real,Q: fun_real_real,R: real] : hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Na,axiom,
    ! [P: fun_complex_complex,Q: fun_nat_complex,R: nat] : hAPP_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,P),Q),R) = hAPP_complex_complex(P,hAPP_nat_complex(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_int_int,R: int] : hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_487,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_nat_int,R: nat] : hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__Rat__Orat_J,axiom,
    ! [P: fun_rat_fun_rat_rat,Q: fun_nat_rat,R: nat] : hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),Q),R) = hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin,axiom,
    ! [P: fun_int_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),Q),R) = hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
    ! [P: fun_in2003698982nt_int,Q: fun_int_int,R: int] : hAPP_i1524277240nt_int(hAPP_f883232462nt_int(hAPP_f2002277285nt_int(cOMBB_1846624359nt_int,P),Q),R) = hAPP_i1524277240nt_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_rat_fun_rat_bool,Q: fun_int_rat,R: int] : hAPP_i16292313t_bool(hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,P),Q),R) = hAPP_r2115560837t_bool(P,hAPP_int_rat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
    ! [P: fun_in351380826t_bool,Q: int,R: int] : hAPP_i1948725293t_bool(hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,P),Q),R) = hAPP_i1948725293t_bool(hAPP_i345030060t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin,axiom,
    ! [P: fun_in1111845413nt_int,Q: int,R: int] : hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,P),Q),R) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin,axiom,
    ! [P: fun_Pr2134412615nt_int,Q: int,R: product_prod_int_int] : hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin,axiom,
    ! [P: fun_Pr2134412615nt_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_488,axiom,
    ! [P: fun_Pr1826394355nt_rat,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1853363688nt_rat(hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,P),Q),R) = hAPP_int_rat(hAPP_P507092031nt_rat(P,R),hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi,axiom,
    ! [P: fun_int_bool,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,P),Q),R) = hAPP_int_bool(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi,axiom,
    ! [P: fun_rat_bool,Q: fun_Pr1024438428nt_rat,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,P),Q),R) = hAPP_rat_bool(P,hAPP_P1853363688nt_rat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu819804446l_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu425979586l_bool,Q: fun_int_fun_nat_bool,R: int] : hAPP_int_bool(hAPP_f41603166t_bool(hAPP_f389116883t_bool(cOMBB_1593174431ol_int,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in938817141l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_489,axiom,
    ! [P: fun_Pr1149636328t_bool,Q: int,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_490,axiom,
    ! [P: fun_Pr1149636328t_bool,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f1325804733t_bool(cOMBS_480885903t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__prod_Itc__Int__O,axiom,
    ! [P: fun_bool_bool,Q: fun_Pr974702441t_bool,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,P),Q),R) = hAPP_bool_bool(P,hAPP_P603027463t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_int_bool,R: int] : hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_491,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obo,axiom,
    ! [P: fun_Pr189909033l_bool,Q: fun_Pr974702441t_bool,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,P),Q),R) = hAPP_bool_bool(hAPP_P921862901l_bool(P,R),hAPP_P603027463t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__RealDef_,axiom,
    ! [P: fun_fu95670548l_real,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,P),Q),R) = hAPP_f1406902706l_real(P,hAPP_n1699378549t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo_492,axiom,
    ! [P: fun_fu819804446l_bool,Q: fun_re546217968t_bool,R: real] : hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_493,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_real_bool,R: real] : hAPP_r1938957037l_bool(hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_real_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDe,axiom,
    ! [P: fun_na182379510l_real,Q: fun_nat_nat,R: nat] : hAPP_n546711566l_real(hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,P),Q),R) = hAPP_n546711566l_real(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H,axiom,
    ! [P: fun_re413263731l_bool,Q: fun_int_real,R: int] : hAPP_i688218208l_bool(hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_int_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O,axiom,
    ! [P: fun_re1879328022l_real,Q: fun_nat_bool,R: real] : hAPP_real_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,P),Q),R) = hAPP_f1406902706l_real(hAPP_r1409264866l_real(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__RealDe,axiom,
    ! [P: fun_re1927296998t_real,Q: fun_real_real,R: real] : hAPP_r474017924t_real(hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,P),Q),R) = hAPP_r474017924t_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
    ! [P: fun_re229243589l_real,Q: fun_int_real,R: int] : hAPP_i371647666l_real(hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_int_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_494,axiom,
    ! [P: fun_re229243589l_real,Q: fun_nat_real,R: nat] : hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__Real,axiom,
    ! [P: fun_fu1122561126l_real,Q: fun_re1927296998t_real,R: real] : hAPP_real_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,P),Q),R) = hAPP_f352196356l_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real,axiom,
    ! [P: fun_na768876520l_real,Q: real,R: nat] : hAPP_n546711566l_real(hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real,axiom,
    ! [P: fun_na768876520l_real,Q: fun_nat_real,R: nat] : hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H_495,axiom,
    ! [P: fun_re413263731l_bool,Q: fun_real_real,R: real] : hAPP_r1134773055l_bool(hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc,axiom,
    ! [P: fun_Pr608224961l_real,Q: fun_na1745536976l_real,R: nat] : hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R) = hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_496,axiom,
    ! [P: fun_Pr1127657839l_bool,Q: fun_re785671361l_real,R: real] : hAPP_real_bool(hAPP_f330985925l_bool(hAPP_f940061907l_bool(cOMBB_1795223523l_real,P),Q),R) = hAPP_P1333854989l_bool(P,hAPP_r1195171167l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__C,axiom,
    ! [P: fun_re398676606omplex,Q: fun_nat_real,R: nat] : hAPP_n1983812447omplex(hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,P),Q),R) = hAPP_r265291036omplex(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_497,axiom,
    ! [P: fun_re229243589l_real,Q: fun_real_real,R: real] : hAPP_r1250527377l_real(hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__fun_Itc_,axiom,
    ! [P: fun_re1227808089l_real,Q: real,R: real] : hAPP_r1250527377l_real(hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_r337325687l_real(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__,axiom,
    ! [P: fun_re785671361l_real,Q: fun_real_real,R: real] : hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f800361423l_real(cOMBB_2107848501l_real,P),Q),R) = hAPP_r1195171167l_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc,axiom,
    ! [P: fun_re1550493891l_real,Q: fun_real_real,R: real] : hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f1730463541l_real(cOMBS_511527878l_real,P),Q),R) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(P,R),hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
    ! [P: fun_in1797533524nt_int,Q: fun_int_int,R: int] : hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,P),Q),R) = hAPP_f1356864277nt_int(hAPP_i744251628nt_int(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_498,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P2110489235nt_int(hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J,axiom,
    ! [P: fun_int_fun_int_rat,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P507092031nt_rat(hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,P),Q),R) = hAPP_int_fun_int_rat(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I,axiom,
    ! [P: fun_in1111845413nt_int,Q: fun_int_int,R: int] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,P),Q),R) = hAPP_i1584592887nt_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__prod_Itc__N,axiom,
    ! [P: fun_na257798253at_nat,Q: fun_nat_nat,R: nat] : hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),Q),R) = hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__,axiom,
    ! [P: fun_fu205926450t_bool,Q: fun_int_fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,P),Q),R) = hAPP_f429935748t_bool(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_499,axiom,
    ! [P: fun_in67169585t_bool,Q: fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,P),Q),R) = hAPP_f429935748t_bool(hAPP_i872162435t_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
    ! [P: fun_in865885604nt_int,Q: fun_int_int,R: int] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,P),Q),R) = hAPP_i1584592887nt_int(hAPP_i1778081398nt_int(P,R),hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__500,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P252494982t_bool(hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_nat,R: nat] : hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu554825757t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc,axiom,
    ! [P: fun_Pr241715507nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f151095827nt_int(cOMBB_1389251822nt_int,P),Q),R) = hAPP_P1975530577nt_int(P,hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__501,axiom,
    ! [P: fun_in719570932t_bool,Q: fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in319042876t_bool,Q: fun_nat_bool,R: int] : hAPP_i418383825t_bool(hAPP_f289318786t_bool(hAPP_f729588221t_bool(cOMBC_678732631t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_i1796122964t_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in719570932t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
    ! [P: fun_in989261322nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f713949180nt_int(cOMBS_1891347093nt_int,P),Q),R) = hAPP_P1975530577nt_int(hAPP_i1216375074nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_502,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Pr974702441t_bool,R: product_prod_int_int] : hAPP_P921862901l_bool(hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_P603027463t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_000tc__fun_,axiom,
    ! [P: fun_fu2100483183t_bool,Q: fun_int_fun_int_real,R: int] : hAPP_i1948725293t_bool(hAPP_f351634443t_bool(hAPP_f1576199059t_bool(cOMBB_95956018ol_int,P),Q),R) = hAPP_f429384717t_bool(P,hAPP_i627165055t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun_,axiom,
    ! [P: fun_fu1532576183t_bool,Q: fun_int_fun_nat_real,R: int] : hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),Q),R) = hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Or,axiom,
    ! [P: fun_in1390825880t_real,Q: fun_int_real,R: int] : hAPP_i627165055t_real(hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,P),Q),R) = hAPP_f269654879t_real(hAPP_i1671359920t_real(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Or,axiom,
    ! [P: fun_in464278240t_real,Q: fun_nat_real,R: int] : hAPP_i157317923t_real(hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,P),Q),R) = hAPP_f1888375463t_real(hAPP_i447943416t_real(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__O,axiom,
    ! [P: fun_re573205052t_real,Q: fun_nat_nat,R: real] : hAPP_r474017924t_real(hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,P),Q),R) = hAPP_f1676298106t_real(hAPP_r50993370t_real(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__503,axiom,
    ! [P: fun_fu1569755401t_real,Q: fun_re1927296998t_real,R: real] : hAPP_r474017924t_real(hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,P),Q),R) = hAPP_f1888375463t_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_I,axiom,
    ! [P: fun_bo1998411783l_real,Q: fun_nat_bool,R: nat] : hAPP_n682674106l_real(hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,P),Q),R) = hAPP_b646336293l_real(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p,axiom,
    ! [P: fun_re1550493891l_real,Q: fun_real_real,R: real] : hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f2114902373l_real(cOMBB_237479429l_real,P),Q),R) = hAPP_r2019696015l_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___504,axiom,
    ! [P: fun_fu215647537t_bool,Q: fun_int_fun_int_int,R: int] : hAPP_i345030060t_bool(hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,P),Q),R) = hAPP_f1791153283t_bool(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_505,axiom,
    ! [P: fun_in2078400507nt_int,Q: fun_int_int,R: int] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,P),Q),R) = hAPP_f883232462nt_int(hAPP_i998473677nt_int(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__,axiom,
    ! [P: fun_re923322561l_real,Q: fun_re785671361l_real,R: real] : hAPP_r1195171167l_real(hAPP_f1736103445l_real(hAPP_f1908301369l_real(cOMBS_902912708l_real,P),Q),R) = hAPP_P1860904029l_real(hAPP_r938550413l_real(P,R),hAPP_r1195171167l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__I,axiom,
    ! [P: fun_in1211676399nt_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,P),Q),R) = hAPP_i61245889nt_int(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__506,axiom,
    ! [P: fun_fu610077323l_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R) = hAPP_f2144054103l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu2104074051l_bool,Q: fun_int_fun_nat_bool,R: int] : hAPP_i614188481l_bool(hAPP_f252743847l_bool(hAPP_f643658017l_bool(cOMBB_2046938128ol_int,P),Q),R) = hAPP_f1146629647l_bool(P,hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL_,axiom,
    ! [P: fun_fu1221449756t_bool,Q: fun_re386138607t_bool,R: real] : hAPP_r481142414t_bool(hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,P),Q),R) = hAPP_f2119767738t_bool(P,hAPP_r632163725t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL_,axiom,
    ! [P: fun_fu1290768832t_bool,Q: fun_re2063775123t_bool,R: real] : hAPP_r481142414t_bool(hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,P),Q),R) = hAPP_f41603166t_bool(P,hAPP_r373117745t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__HOL__O,axiom,
    ! [P: fun_re1058169246t_bool,Q: fun_int_bool,R: real] : hAPP_r632163725t_bool(hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,P),Q),R) = hAPP_f1048215610t_bool(hAPP_r1623423338t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O_507,axiom,
    ! [P: fun_re507499238t_bool,Q: fun_nat_bool,R: real] : hAPP_r373117745t_bool(hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,P),Q),R) = hAPP_f289318786t_bool(hAPP_r653725362t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__508,axiom,
    ! [P: fun_fu705349683l_real,Q: fun_re1927296998t_real,R: real] : hAPP_r1409264866l_real(hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,P),Q),R) = hAPP_f1509287679l_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___509,axiom,
    ! [P: fun_fu1480724285nt_int,Q: fun_int_fun_int_int,R: int] : hAPP_i744251628nt_int(hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,P),Q),R) = hAPP_f747162069nt_int(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___510,axiom,
    ! [P: fun_fu1843361787nt_int,Q: fun_int_fun_int_int,R: int] : hAPP_i1778081398nt_int(hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,P),Q),R) = hAPP_f1574055885nt_int(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea,axiom,
    ! [P: fun_fu1280874925l_bool,Q: fun_re1550493891l_real,R: real] : hAPP_r1134773055l_bool(hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,P),Q),R) = hAPP_f330985925l_bool(P,hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__511,axiom,
    ! [P: fun_fu2112913523t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i872162435t_bool(hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,P),Q),R) = hAPP_f1512942609t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__512,axiom,
    ! [P: fun_fu257678363t_bool,Q: fun_in351380826t_bool,R: int] : hAPP_i345030060t_bool(hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,P),Q),R) = hAPP_f428220345t_bool(P,hAPP_i345030060t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I,axiom,
    ! [P: fun_in792564872nt_int,Q: fun_in1111845413nt_int,R: int] : hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R) = hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
    ! [P: fun_in941977532nt_int,Q: product_prod_int_int,R: int] : hAPP_i1216375074nt_int(hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint__513,axiom,
    ! [P: fun_in941977532nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1216375074nt_int(hAPP_f1375227553nt_int(hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc,axiom,
    ! [P: fun_fu1909738875t_bool,Q: fun_in531499254l_bool,R: int] : hAPP_i1529485324t_bool(hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,P),Q),R) = hAPP_f627970963t_bool(P,hAPP_i68813070l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_,axiom,
    ! [P: fun_fu2120820641l_real,Q: fun_re229243589l_real,R: real] : hAPP_r337325687l_real(hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,P),Q),R) = hAPP_f556082547l_real(P,hAPP_r1250527377l_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
    ! [P: fun_re955532029l_real,Q: fun_real_real,R: real] : hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
    ! [P: fun_re955532029l_real,Q: fun_re229243589l_real,R: real] : hAPP_r2019696015l_real(hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),hAPP_r1250527377l_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It,axiom,
    ! [P: fun_re521366895t_bool,Q: fun_int_fun_int_real,R: real] : hAPP_r632163725t_bool(hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,P),Q),R) = hAPP_f351634443t_bool(hAPP_r1392521531t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It_514,axiom,
    ! [P: fun_re811280567t_bool,Q: fun_int_fun_nat_real,R: real] : hAPP_r373117745t_bool(hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,P),Q),R) = hAPP_f1222178131t_bool(hAPP_r2052264067t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__515,axiom,
    ! [P: fun_fu1719442527t_real,Q: fun_re1927296998t_real,R: real] : hAPP_r50993370t_real(hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,P),Q),R) = hAPP_f1566856189t_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun,axiom,
    ! [P: fun_fu1716408315t_bool,Q: fun_re413263731l_bool,R: real] : hAPP_r847664445t_bool(hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,P),Q),R) = hAPP_f1173902867t_bool(P,hAPP_r1134773055l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_516,axiom,
    ! [P: fun_fu1635948867t_bool,Q: fun_re413263731l_bool,R: real] : hAPP_r279757445t_bool(hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,P),Q),R) = hAPP_f605995867t_bool(P,hAPP_r1134773055l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__517,axiom,
    ! [P: fun_fu2064567419t_real,Q: fun_in7315610l_real,R: int] : hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R) = hAPP_f766218899t_real(P,hAPP_i371647666l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__518,axiom,
    ! [P: fun_fu1138019779t_real,Q: fun_in7315610l_real,R: int] : hAPP_i447943416t_real(hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,P),Q),R) = hAPP_f1690286043t_real(P,hAPP_i371647666l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
    ! [P: fun_Pr1238001172nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1153444985nt_int(hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,P),Q),R) = hAPP_P621635040nt_int(P,hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
    ! [P: fun_Pr642042703nt_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,P),Q),R) = hAPP_f521271395nt_int(hAPP_P752050971nt_int(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__,axiom,
    ! [P: fun_bo1552468991nt_int,Q: fun_int_bool,R: int] : hAPP_i2104874254nt_int(hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,P),Q),R) = hAPP_b40753821nt_int(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,axiom,
    ! [P: fun_fu587508068t_bool,Q: fun_in397122125l_bool,R: int] : hAPP_i1529485324t_bool(hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,P),Q),R) = hAPP_f727283836t_bool(P,hAPP_i1823918693l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,axiom,
    ! [P: fun_fu1901564496t_bool,Q: fun_in1678813097l_bool,R: int] : hAPP_i1796122964t_bool(hAPP_f1902329768t_bool(hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,P),Q),R) = hAPP_f561022312t_bool(P,hAPP_i614188481l_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
    ! [P: fun_Pr755126262t_bool,Q: fun_Pr2144253564nt_int,R: product_prod_int_int] : hAPP_P989584703t_bool(hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,P),Q),R) = hAPP_f892584630t_bool(hAPP_P1400155668t_bool(P,R),hAPP_P1668131738nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__H,axiom,
    ! [P: fun_fu2112443261t_bool,Q: fun_Pr313471731t_bool,R: product_prod_int_int] : hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,P),Q),R) = hAPP_f237669397t_bool(P,hAPP_P989584703t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__519,axiom,
    ! [P: fun_Pr1254492423t_bool,Q: fun_Pr313471731t_bool,R: product_prod_int_int] : hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,P),Q),R) = hAPP_f237669397t_bool(hAPP_P877703507t_bool(P,R),hAPP_P989584703t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__520,axiom,
    ! [P: fun_fu1230750243l_bool,Q: fun_re386138607t_bool,R: real] : hAPP_r1150855451l_bool(hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,P),Q),R) = hAPP_f1344929775l_bool(P,hAPP_r632163725t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__521,axiom,
    ! [P: fun_fu970548443l_bool,Q: fun_re2063775123t_bool,R: real] : hAPP_r285062775l_bool(hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,P),Q),R) = hAPP_f252743847l_bool(P,hAPP_r373117745t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal,axiom,
    ! [P: fun_fu888010351l_real,Q: fun_re1227808089l_real,R: real] : hAPP_r337325687l_real(hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,P),Q),R) = hAPP_f203520653l_real(P,hAPP_r337325687l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__522,axiom,
    ! [P: fun_fu1950887583l_real,Q: fun_re229243589l_real,R: real] : hAPP_r579619957l_real(hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,P),Q),R) = hAPP_f1591894897l_real(P,hAPP_r1250527377l_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p,axiom,
    ! [P: fun_re1233422975l_real,Q: fun_re1550493891l_real,R: real] : hAPP_r2019696015l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,P),Q),R) = hAPP_f1736103445l_real(hAPP_r224952523l_real(P,R),hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc___523,axiom,
    ! [P: fun_re1761459671l_real,Q: fun_re785671361l_real,R: real] : hAPP_r938550413l_real(hAPP_f1293550189l_real(hAPP_f183342685l_real(cOMBS_794971624l_real,P),Q),R) = hAPP_P1982799375l_real(hAPP_r931665397l_real(P,R),hAPP_r1195171167l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod,axiom,
    ! [P: fun_fu2104383023nt_int,Q: fun_in865885604nt_int,R: int] : hAPP_i1778081398nt_int(hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,P),Q),R) = hAPP_f465821005nt_int(P,hAPP_i1778081398nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int,axiom,
    ! [P: fun_fu156492935nt_int,Q: fun_in1111845413nt_int,R: int] : hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),Q),R) = hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int_524,axiom,
    ! [P: fun_fu1379407829nt_int,Q: fun_in1111845413nt_int,R: int] : hAPP_i1153444985nt_int(hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,P),Q),R) = hAPP_f1375227553nt_int(P,hAPP_i1584592887nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__I,axiom,
    ! [P: fun_fu686082665t_bool,Q: fun_Pr2144253564nt_int,R: product_prod_int_int] : hAPP_P2053772734t_bool(hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,P),Q),R) = hAPP_f291087797t_bool(P,hAPP_P1668131738nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_,axiom,
    ! [P: fun_bo1189210245l_real,Q: fun_real_bool,R: real] : hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R) = hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___525,axiom,
    ! [P: fun_fu2013528197nt_int,Q: fun_Pr2134412615nt_int,R: product_prod_int_int] : hAPP_P752050971nt_int(hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,P),Q),R) = hAPP_f11151005nt_int(P,hAPP_P2110489235nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc_526,axiom,
    ! [P: fun_fu201758811t_bool,Q: fun_Pr189909033l_bool,R: product_prod_int_int] : hAPP_P877703507t_bool(hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,P),Q),R) = hAPP_f513299955t_bool(P,hAPP_P921862901l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_,axiom,
    ! [P: fun_fu1269744292t_bool,Q: fun_re1598305103l_bool,R: real] : hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),Q),R) = hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__fun_,axiom,
    ! [P: fun_fu553683856t_bool,Q: fun_re1564496555l_bool,R: real] : hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R) = hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea_527,axiom,
    ! [P: fun_fu383125559l_real,Q: fun_re1550493891l_real,R: real] : hAPP_r712953929l_real(hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,P),Q),R) = hAPP_f800361423l_real(P,hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p_528,axiom,
    ! [P: fun_re1414183599l_real,Q: fun_re1550493891l_real,R: real] : hAPP_r1230691443l_real(hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,P),Q),R) = hAPP_f1293550189l_real(hAPP_r1043989325l_real(P,R),hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p,axiom,
    ! [P: fun_fu2102390523nt_int,Q: fun_in989261322nt_int,R: int] : hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),Q),R) = hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_,axiom,
    ! [P: fun_fu407119589t_bool,Q: fun_re1979725814t_bool,R: real] : hAPP_r1623423338t_bool(hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,P),Q),R) = hAPP_f472159229t_bool(P,hAPP_r1962736578t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_,axiom,
    ! [P: fun_fu1083576549t_bool,Q: fun_re506364222t_bool,R: real] : hAPP_r653725362t_bool(hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,P),Q),R) = hAPP_f729588221t_bool(P,hAPP_r1562208522t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mt,axiom,
    ! [P: fun_fu916648827t_bool,Q: fun_re1171866993t_bool,R: real] : hAPP_r1392521531t_bool(hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,P),Q),R) = hAPP_f1576199059t_bool(P,hAPP_r847664445t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mt,axiom,
    ! [P: fun_fu1861895035t_bool,Q: fun_re1091407545t_bool,R: real] : hAPP_r2052264067t_bool(hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,P),Q),R) = hAPP_f743790483t_bool(P,hAPP_r279757445t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_529,axiom,
    ! [P: fun_fu380422627l_real,Q: fun_re413263731l_bool,R: real] : hAPP_r794344869l_real(hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,P),Q),R) = hAPP_f1607917947l_real(P,hAPP_r1134773055l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_530,axiom,
    ! [P: fun_fu1698495721l_real,Q: fun_re665330775l_real,R: real] : hAPP_r712953929l_real(hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,P),Q),R) = hAPP_f1730463541l_real(P,hAPP_r579619957l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int__Oint,axiom,
    ! [P: fun_fu1280007140nt_int,Q: fun_in2124216673nt_int,R: int] : hAPP_i1836591008nt_int(hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,P),Q),R) = hAPP_f713949180nt_int(P,hAPP_i1153444985nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__f,axiom,
    ! [P: fun_fu581830891t_bool,Q: fun_Pr820506994t_bool,R: product_prod_int_int] : hAPP_P1400155668t_bool(hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,P),Q),R) = hAPP_f1325804733t_bool(P,hAPP_P2053772734t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real,axiom,
    ! [P: fun_fu2079587053l_real,Q: fun_re10870101l_real,R: real] : hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),Q),R) = hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real_531,axiom,
    ! [P: fun_fu209750539l_real,Q: fun_re69907033l_real,R: real] : hAPP_r1043989325l_real(hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,P),Q),R) = hAPP_f183342685l_real(P,hAPP_r794344869l_real(Q,R)) ).

%----Conjectures (1)
% tff(conj_0,conjecture,
%     hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,a),b))),hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,p),q))) = hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,a),p)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,b),q))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,a),q)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,b),p)))) ).

%------------------------------------------------------------------------------